In this study, a standard cubic congruence of composite modulus a multiple of four and power of three is considered. After a thorough study, formulation of the solutions of the said congruence are established in four different cases. It is found that the standard cubic congruence under consideration in case I has exactly three solutions for an odd positive integer and in case II, has six solutions for an even positive integer. In case III, it has nine solutions and in case IV, the congruence has exactly eighteen solutions. Formulations are the merit of the study. Prof B M Roy "Rp-72: Formulation of Solutions of a Standard Cubic Congruence of Composite Modulus-a Multiple of Four and Power of Three" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-3 , April 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30762.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/30762/rp72-formulation-of-solutions-of-a-standard-cubic-congruence-of-composite-modulusa-multiple-of-four-and-power-of-three/prof-b-m-roy
RP 129 Formulation of a Special Type of Standard Quadratic Congruence of Comp...ijtsrd
In this current study, a very special type of standard quadratic congruence of composite modulus a product of two powered odd primes is considered for finding solutions. It is studied and solved in different cases. The author's effort made an easy way to find the solutions of the congruence under consideration. Such types of congruence are not formulated earlier. First time the author's effort came in existence. This is the merit of the paper. Prof B M Roy "RP-129: Formulation of a Special Type of Standard Quadratic Congruence of Composite Modulus- a Product of Two Powered Odd Primes" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30964.pdf Paper Url :https://www.ijtsrd.com/mathemetics/applied-mathematics/30964/rp129-formulation-of-a-special-type-of-standard-quadratic-congruence-of-composite-modulus%E2%80%93-a-product-of-two-powered-odd-primes/prof-b-m-roy
Rp-99: Formulation of Standard Cubic Congruence of Composite Modulus- A Multi...ijtsrd
In this paper, two special standard cubic congruence of composite modulus of higher degree are considered for study to establish a formulation of the solutions of the congruence. The formulation of the solutions is established. The author's rigorous labour works efficiently and succeed to formulate the solutions of the congruence under consideration. This paper presents the author's successful efforts. The discovered formula is tested and verified true by illustrating different suitable examples. No effective method is found in the literature of mathematics for its solutions. Formulation is the merit of the paper. Formulation is proved time saving, easy and simple. The study of standard cubic congruence of composite modulus of higher degree is now an interesting subject of mathematics. Prof B M Roy "Rp-99: Formulation of Standard Cubic Congruence of Composite Modulus- A Multiple of the Power of the Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-1 , December 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29391.pdfPaper URL: https://www.ijtsrd.com/mathemetics/other/29391/rp-99-formulation-of-standard-cubic-congruence-of-composite-modulus--a-multiple-of-the-power-of-the-modulus/prof-b-m-roy
RP 64 A Review of Finding Solutions of Standard Cubic Congruence of Prime Mod...ijtsrd
In this paper, the solutions of a standard cubic congruence of prime modulus is discussed. Also, the condition of solvability is established. The literature of mathematics is silent for finding the solutions of such congruence. Even no method to solve such types of congruence is discussed. Here, solvability condition is obtained. The solvability condition makes it easy to find the solutions. It saves the time in calculation. This is the merit of the paper. Prof B M Roy "RP-64: A Review of Finding Solutions of Standard Cubic Congruence of Prime Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-3 , April 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30421.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/30421/rp64-a-review-of-finding-solutions-of-standard-cubic-congruence-of-prime-modulus/prof-b-m-roy
Rp-106: Formulation of Solutions of a Class of Standard Congruence of Composi...ijtsrd
In this paper, a standard congruence of higher degree of composite modulus is considered for study and after a thorough study, it is then formulated for its incongruent solutions. The established formula is tested and verified true by solving suitable examples. Formulation is the merit of the paper. This formulation made the study of the congruence of higher degree easy, simple and interesting. Mental calculation of solutions become possible due to this formulation. In the literature of earlier mathematics, no formulation is found for the solutions of the said congruence. Prof B M Roy "Rp-106: Formulation of Solutions of a Class of Standard Congruence of Composite Modulus of Higher Degree" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-1 , December 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29764.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29764/rp-106-formulation-of-solutions-of-a-class-of-standard-congruence-of-composite-modulus-of-higher-degree/prof-b-m-roy
RP 47 Formulation of a Class of Standard Congruence of Higher Degree Modulo a...ijtsrd
In this paper, the author considered a problem ""Formulation of solutions of a class of standard congruence of higher degree modulo a power of an odd prime integer"". It is studied rigorously and the solutions of the congruence are formulated. Illustrating suitable examples, the formula established is tested and verified true. No earlier formulation is found in the literature of mathematics for such congruence. Author has tried his best to find a formula for the solutions of the problem under consideration and succeed. It is the merit of the paper. Prof B M Roy ""RP-47: Formulation of a Class of Standard Congruence of Higher Degree Modulo a Power of an Odd Prime Integer"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020,
URL: https://www.ijtsrd.com/papers/ijtsrd29994.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/other/29994/rp-47-formulation-of-a-class-of-standard-congruence-of-higher-degree-modulo-a-power-of-an-odd-prime-integer/prof-b-m-roy
RP 144 Formulation of Solutions of a Very Special Class of Standard Quadratic...ijtsrd
In this paper, the author considered a very special type of standard quadratic congruence of composite modulus for its formulation of solutions. It is found that such a congruence has exactly 4p solutions, p being an odd prime present in the modulus of the congruence. These solutions are formulated by the author. First time a formula is derived and hence the solutions can be obtained orally. The literature of mathematics remains silent in case of formulation of such congruence. Formulation is the merit of the paper. Prof B M Roy "RP-144: Formulation of Solutions of a Very Special Class of Standard Quadratic Congruence of Composite Modulus modulo a Multiple of Powered Even Prime by a Powered Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd35718.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/35718/rp144-formulation-of-solutions-of-a-very-special-class-of-standard-quadratic-congruence-of-composite-modulus-modulo-a-multiple-of-powered-even-prime-by-a-powered-odd-prime/prof-b-m-roy
RP 129 Formulation of a Special Type of Standard Quadratic Congruence of Comp...ijtsrd
In this current study, a very special type of standard quadratic congruence of composite modulus a product of two powered odd primes is considered for finding solutions. It is studied and solved in different cases. The author's effort made an easy way to find the solutions of the congruence under consideration. Such types of congruence are not formulated earlier. First time the author's effort came in existence. This is the merit of the paper. Prof B M Roy "RP-129: Formulation of a Special Type of Standard Quadratic Congruence of Composite Modulus- a Product of Two Powered Odd Primes" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30964.pdf Paper Url :https://www.ijtsrd.com/mathemetics/applied-mathematics/30964/rp129-formulation-of-a-special-type-of-standard-quadratic-congruence-of-composite-modulus%E2%80%93-a-product-of-two-powered-odd-primes/prof-b-m-roy
Rp-99: Formulation of Standard Cubic Congruence of Composite Modulus- A Multi...ijtsrd
In this paper, two special standard cubic congruence of composite modulus of higher degree are considered for study to establish a formulation of the solutions of the congruence. The formulation of the solutions is established. The author's rigorous labour works efficiently and succeed to formulate the solutions of the congruence under consideration. This paper presents the author's successful efforts. The discovered formula is tested and verified true by illustrating different suitable examples. No effective method is found in the literature of mathematics for its solutions. Formulation is the merit of the paper. Formulation is proved time saving, easy and simple. The study of standard cubic congruence of composite modulus of higher degree is now an interesting subject of mathematics. Prof B M Roy "Rp-99: Formulation of Standard Cubic Congruence of Composite Modulus- A Multiple of the Power of the Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-1 , December 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29391.pdfPaper URL: https://www.ijtsrd.com/mathemetics/other/29391/rp-99-formulation-of-standard-cubic-congruence-of-composite-modulus--a-multiple-of-the-power-of-the-modulus/prof-b-m-roy
RP 64 A Review of Finding Solutions of Standard Cubic Congruence of Prime Mod...ijtsrd
In this paper, the solutions of a standard cubic congruence of prime modulus is discussed. Also, the condition of solvability is established. The literature of mathematics is silent for finding the solutions of such congruence. Even no method to solve such types of congruence is discussed. Here, solvability condition is obtained. The solvability condition makes it easy to find the solutions. It saves the time in calculation. This is the merit of the paper. Prof B M Roy "RP-64: A Review of Finding Solutions of Standard Cubic Congruence of Prime Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-3 , April 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30421.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/30421/rp64-a-review-of-finding-solutions-of-standard-cubic-congruence-of-prime-modulus/prof-b-m-roy
Rp-106: Formulation of Solutions of a Class of Standard Congruence of Composi...ijtsrd
In this paper, a standard congruence of higher degree of composite modulus is considered for study and after a thorough study, it is then formulated for its incongruent solutions. The established formula is tested and verified true by solving suitable examples. Formulation is the merit of the paper. This formulation made the study of the congruence of higher degree easy, simple and interesting. Mental calculation of solutions become possible due to this formulation. In the literature of earlier mathematics, no formulation is found for the solutions of the said congruence. Prof B M Roy "Rp-106: Formulation of Solutions of a Class of Standard Congruence of Composite Modulus of Higher Degree" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-1 , December 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29764.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29764/rp-106-formulation-of-solutions-of-a-class-of-standard-congruence-of-composite-modulus-of-higher-degree/prof-b-m-roy
RP 47 Formulation of a Class of Standard Congruence of Higher Degree Modulo a...ijtsrd
In this paper, the author considered a problem ""Formulation of solutions of a class of standard congruence of higher degree modulo a power of an odd prime integer"". It is studied rigorously and the solutions of the congruence are formulated. Illustrating suitable examples, the formula established is tested and verified true. No earlier formulation is found in the literature of mathematics for such congruence. Author has tried his best to find a formula for the solutions of the problem under consideration and succeed. It is the merit of the paper. Prof B M Roy ""RP-47: Formulation of a Class of Standard Congruence of Higher Degree Modulo a Power of an Odd Prime Integer"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020,
URL: https://www.ijtsrd.com/papers/ijtsrd29994.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/other/29994/rp-47-formulation-of-a-class-of-standard-congruence-of-higher-degree-modulo-a-power-of-an-odd-prime-integer/prof-b-m-roy
RP 144 Formulation of Solutions of a Very Special Class of Standard Quadratic...ijtsrd
In this paper, the author considered a very special type of standard quadratic congruence of composite modulus for its formulation of solutions. It is found that such a congruence has exactly 4p solutions, p being an odd prime present in the modulus of the congruence. These solutions are formulated by the author. First time a formula is derived and hence the solutions can be obtained orally. The literature of mathematics remains silent in case of formulation of such congruence. Formulation is the merit of the paper. Prof B M Roy "RP-144: Formulation of Solutions of a Very Special Class of Standard Quadratic Congruence of Composite Modulus modulo a Multiple of Powered Even Prime by a Powered Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd35718.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/35718/rp144-formulation-of-solutions-of-a-very-special-class-of-standard-quadratic-congruence-of-composite-modulus-modulo-a-multiple-of-powered-even-prime-by-a-powered-odd-prime/prof-b-m-roy
RP 108 Formulation of Solutions of Standard Congruence of Higher Degree modul...ijtsrd
In this study, the solutions of a standard congruence of higher degree modulo a multiple of composite power integer is considered for formulation and is formulated. The formula developed is proved and verified true. Oral calculation of solutions are also possible. Formulation is the merit of the paper. Prof B M Roy ""RP-108: Formulation of Solutions of Standard Congruence of Higher Degree modulo a Multiple of Composite Power Integer"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020,
URL: https://www.ijtsrd.com/papers/ijtsrd30151.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/other/30151/rp-108-formulation-of-solutions-of-standard-congruence-of-higher-degree-modulo-a-multiple-of-composite-power-integer/prof-b-m-roy
RP 166 Solving Some Special Standard Cubic Congruence of Composite Modulus mo...ijtsrd
Here in this paper, ten special type of standard cubic congruence of composite modulus are studied for their solutions. It is found that each of the cubic congruence under consideration has a single solution. The solution can be obtained orally as the solution is given in the problems. No extra effort is necessary to find the solution. Prof B M Roy "RP-166: Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-4 , June 2021, URL: https://www.ijtsrd.compapers/ijtsrd42321.pdf Paper URL: https://www.ijtsrd.commathemetics/applied-mathamatics/42321/rp166-solving-some-special-standard-cubic-congruence-of-composite-modulus-modulo-a-multiple-of-an-odd-prime/prof-b-m-roy
RP 105 Formulation of Standard Quadratic Congruence of Composite Modulus A Pr...ijtsrd
In this paper, a standard quadratic congruence of composite modulus a product of twin primes, is formulated. Actually a formula is discovered to find all the solutions of the congruence. It is found that the method is simple and takes less time as compared to the existed method. A comparative study is made by solving a problem using existed method and Author's formulation. Formulation is proved time saving. Solutions can be obtained orally. This is the merit of the paper. Prof B M Roy "RP-105: Formulation of Standard Quadratic Congruence of Composite Modulus- A Product of Twin Primes" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26492.pdfPaper URL: https://www.ijtsrd.com/mathemetics/other/26492/rp-105-formulation-of-standard-quadratic-congruence-of-composite-modulus--a-product-of-twin-primes/prof-b-m-roy
RP 74 Solving Three Special Types of Standard Congruence of Prime Modulus of ...ijtsrd
In this paper, three special classes of congruence of prime modulus of higher degree are considered for formulation and authors efforts established the formulae for solutions. The congruence were not formulated by earlier mathematicians. Due to this formulation, it becomes an easy task to solve the congruence of higher degree of prime modulus reducing the congruence to their equivalent quadratic congruence of prime modulus. Formulations of the congruence is the merit of the paper. Prof. B M Roy ""RP-74: Solving Three Special Types of Standard Congruence of Prime Modulus of Higher Degree"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23488.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/23488/rp-74-solving-three-special-types-of-standard-congruence-of-prime-modulus-of-higher-degree/prof-b-m-roy
RP 170 Formulating Solutions of a Special Class of Standard Cubic Congruence ...ijtsrd
In this research paper, the author has formulated the solutions of a standard cubic congruence of composite modulus modulo p2 multiple of 3n, n being a positive integer. It is found that the said congruence has exactly 3p incongruent solutions p being an odd prime positive integer. It seems that the earlier mathematicians didn’t study such types of congruence. First time a formulation of solutions is established by the author. Formulation is the merit of the paper as formulation of solutions has provided a simple procedure of finding the required solutions with an ease. Prof B M Roy "RP-170: Formulating Solutions of a Special Class of Standard Cubic Congruence Modulo p2 Multiple of 3n" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-4 , June 2021, URL: https://www.ijtsrd.compapers/ijtsrd42540.pdf Paper URL: https://www.ijtsrd.commathemetics/applied-mathamatics/42540/rp170-formulating-solutions-of-a-special-class-of-standard-cubic-congruence-modulo-p2-multiple-of-3n/prof-b-m-roy
RP 75 Formulation of two Special Classes of Standard Congruence of Prime Modu...ijtsrd
In this paper, two special classes of congruence of prime modulus of higher degree are considered for formulation. Authors sincere efforts established the corresponding formulae for solutions. The congruence were not formulated earlier. Due to this formulation, it becomes an easy task to solve the said congruence orally when p is comparatively small. Formulations of the congruence is the merit of the paper. Prof. B M Roy ""RP-75: Formulation of two Special Classes of Standard Congruence of Prime Modulus of Higher Degree"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23417.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/23417/rp-75-formulation-of-two-special-classes-of-standard-congruence-of-prime-modulus-of-higher-degree/prof-b-m-roy
RP 163 Solving a Special Standard Quadratic Congruence Modulo an Even Multipl...YogeshIJTSRD
The author, here in this paper, presented a formulation for solving a special standard quadratic congruence modulo an even multiple of an odd positive integer. The established formula is tested and verified true by solving various numerical examples. The formulation works well and proved time saving. Prof B M Roy "RP-163: Solving a Special Standard Quadratic Congruence Modulo an Even Multiple of an Odd Positive Integer" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd41116.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/41116/rp163-solving-a-special-standard-quadratic-congruence-modulo-an-even-multiple-of-an-odd-positive-integer/prof-b-m-roy
Rp 91 Solving Some Standard Cubic Congruence of Prime Modulusijtsrd
In this paper, finding solutions of some classes of standard cubic congruence of prime modulus are considered for study and then formulated. Formulation of the solutions is proved time saving, simple and quick. It made the finding solutions of cubic congruence of prime modulus easy. Formulation is the merit of the paper. . Here, solvability condition is obtained. This saves time in calculation. This is the merit of the paper. Prof B M Roy ""Rp-91: Solving Some Standard Cubic Congruence of Prime Modulus"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25117.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25117/rp-91--solving-some-standard-cubic-congruence-of-prime-modulus/prof-b-m-roy
A Review Article on Fixed Point Theory and Its Applicationijtsrd
The theory of fixed point is one of the most important and powerful tools of the modern mathematics not only it is used on a daily bases in pure and applied mathematics but it is also solving a bridge between analysis and topology and provide a very fruitful are of interaction between the two. The theory of fixed points belongs to topology, a part of mathematics created at the end of the nineteenth century. The famous French mathematician H. Poincare 1854 1912 was the founder of the fixed point approach. He had deep insight into its future importance for problems of mathematical analysis and celestial mechanics and took an active part in its development. Dr. Brajraj Singh Chauhan "A Review Article on Fixed Point Theory & Its Application" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26431.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/26431/a-review-article-on-fixed-point-theory-and-its-application/dr-brajraj-singh-chauhan
Lecture #2 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
Lecture #3 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
RP 165 Solving Some Special Classes of Standard Cubic Congruence of Composite...YogeshIJTSRD
In this paper, a special class of standard cubic congruence of composite modulus is studied and formulated the solutions. The discussion is presented here. This cubic congruence has three types of solutions. In first case, it has exactly 3 incongruent solutions in second case, it has nine incongruent solutions and in third case it has twenty seven incongruent solutions. First time a suitable formulation is available for a class of standard cubic congruence. So, formulation is the merit of the paper. Prof B M Roy "RP-165: Solving Some Special Classes of Standard Cubic Congruence of Composite Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd39816.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/39816/rp165-solving-some-special-classes-of-standard-cubic-congruence-of-composite-modulus/prof-b-m-roy
Finding the Extreme Values with some Application of Derivativesijtsrd
There are many different way of mathematics rules. Among them, we express finding the extreme values for the optimization problems that changes in the particle life with the derivatives. The derivative is the exact rate at which one quantity changes with respect to another. And them, we can compute the profit and loss of a process that a company or a system. Variety of optimization problems are solved by using derivatives. There were use derivatives to find the extreme values of functions, to determine and analyze the shape of graphs and to find numerically where a function equals zero. Kyi Sint | Kay Thi Win "Finding the Extreme Values with some Application of Derivatives" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29347.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29347/finding-the-extreme-values-with-some-application-of-derivatives/kyi-sint
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
RP 108 Formulation of Solutions of Standard Congruence of Higher Degree modul...ijtsrd
In this study, the solutions of a standard congruence of higher degree modulo a multiple of composite power integer is considered for formulation and is formulated. The formula developed is proved and verified true. Oral calculation of solutions are also possible. Formulation is the merit of the paper. Prof B M Roy ""RP-108: Formulation of Solutions of Standard Congruence of Higher Degree modulo a Multiple of Composite Power Integer"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020,
URL: https://www.ijtsrd.com/papers/ijtsrd30151.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/other/30151/rp-108-formulation-of-solutions-of-standard-congruence-of-higher-degree-modulo-a-multiple-of-composite-power-integer/prof-b-m-roy
RP 166 Solving Some Special Standard Cubic Congruence of Composite Modulus mo...ijtsrd
Here in this paper, ten special type of standard cubic congruence of composite modulus are studied for their solutions. It is found that each of the cubic congruence under consideration has a single solution. The solution can be obtained orally as the solution is given in the problems. No extra effort is necessary to find the solution. Prof B M Roy "RP-166: Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-4 , June 2021, URL: https://www.ijtsrd.compapers/ijtsrd42321.pdf Paper URL: https://www.ijtsrd.commathemetics/applied-mathamatics/42321/rp166-solving-some-special-standard-cubic-congruence-of-composite-modulus-modulo-a-multiple-of-an-odd-prime/prof-b-m-roy
RP 105 Formulation of Standard Quadratic Congruence of Composite Modulus A Pr...ijtsrd
In this paper, a standard quadratic congruence of composite modulus a product of twin primes, is formulated. Actually a formula is discovered to find all the solutions of the congruence. It is found that the method is simple and takes less time as compared to the existed method. A comparative study is made by solving a problem using existed method and Author's formulation. Formulation is proved time saving. Solutions can be obtained orally. This is the merit of the paper. Prof B M Roy "RP-105: Formulation of Standard Quadratic Congruence of Composite Modulus- A Product of Twin Primes" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26492.pdfPaper URL: https://www.ijtsrd.com/mathemetics/other/26492/rp-105-formulation-of-standard-quadratic-congruence-of-composite-modulus--a-product-of-twin-primes/prof-b-m-roy
RP 74 Solving Three Special Types of Standard Congruence of Prime Modulus of ...ijtsrd
In this paper, three special classes of congruence of prime modulus of higher degree are considered for formulation and authors efforts established the formulae for solutions. The congruence were not formulated by earlier mathematicians. Due to this formulation, it becomes an easy task to solve the congruence of higher degree of prime modulus reducing the congruence to their equivalent quadratic congruence of prime modulus. Formulations of the congruence is the merit of the paper. Prof. B M Roy ""RP-74: Solving Three Special Types of Standard Congruence of Prime Modulus of Higher Degree"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23488.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/23488/rp-74-solving-three-special-types-of-standard-congruence-of-prime-modulus-of-higher-degree/prof-b-m-roy
RP 170 Formulating Solutions of a Special Class of Standard Cubic Congruence ...ijtsrd
In this research paper, the author has formulated the solutions of a standard cubic congruence of composite modulus modulo p2 multiple of 3n, n being a positive integer. It is found that the said congruence has exactly 3p incongruent solutions p being an odd prime positive integer. It seems that the earlier mathematicians didn’t study such types of congruence. First time a formulation of solutions is established by the author. Formulation is the merit of the paper as formulation of solutions has provided a simple procedure of finding the required solutions with an ease. Prof B M Roy "RP-170: Formulating Solutions of a Special Class of Standard Cubic Congruence Modulo p2 Multiple of 3n" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-4 , June 2021, URL: https://www.ijtsrd.compapers/ijtsrd42540.pdf Paper URL: https://www.ijtsrd.commathemetics/applied-mathamatics/42540/rp170-formulating-solutions-of-a-special-class-of-standard-cubic-congruence-modulo-p2-multiple-of-3n/prof-b-m-roy
RP 75 Formulation of two Special Classes of Standard Congruence of Prime Modu...ijtsrd
In this paper, two special classes of congruence of prime modulus of higher degree are considered for formulation. Authors sincere efforts established the corresponding formulae for solutions. The congruence were not formulated earlier. Due to this formulation, it becomes an easy task to solve the said congruence orally when p is comparatively small. Formulations of the congruence is the merit of the paper. Prof. B M Roy ""RP-75: Formulation of two Special Classes of Standard Congruence of Prime Modulus of Higher Degree"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23417.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/23417/rp-75-formulation-of-two-special-classes-of-standard-congruence-of-prime-modulus-of-higher-degree/prof-b-m-roy
RP 163 Solving a Special Standard Quadratic Congruence Modulo an Even Multipl...YogeshIJTSRD
The author, here in this paper, presented a formulation for solving a special standard quadratic congruence modulo an even multiple of an odd positive integer. The established formula is tested and verified true by solving various numerical examples. The formulation works well and proved time saving. Prof B M Roy "RP-163: Solving a Special Standard Quadratic Congruence Modulo an Even Multiple of an Odd Positive Integer" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd41116.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/41116/rp163-solving-a-special-standard-quadratic-congruence-modulo-an-even-multiple-of-an-odd-positive-integer/prof-b-m-roy
Rp 91 Solving Some Standard Cubic Congruence of Prime Modulusijtsrd
In this paper, finding solutions of some classes of standard cubic congruence of prime modulus are considered for study and then formulated. Formulation of the solutions is proved time saving, simple and quick. It made the finding solutions of cubic congruence of prime modulus easy. Formulation is the merit of the paper. . Here, solvability condition is obtained. This saves time in calculation. This is the merit of the paper. Prof B M Roy ""Rp-91: Solving Some Standard Cubic Congruence of Prime Modulus"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25117.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25117/rp-91--solving-some-standard-cubic-congruence-of-prime-modulus/prof-b-m-roy
A Review Article on Fixed Point Theory and Its Applicationijtsrd
The theory of fixed point is one of the most important and powerful tools of the modern mathematics not only it is used on a daily bases in pure and applied mathematics but it is also solving a bridge between analysis and topology and provide a very fruitful are of interaction between the two. The theory of fixed points belongs to topology, a part of mathematics created at the end of the nineteenth century. The famous French mathematician H. Poincare 1854 1912 was the founder of the fixed point approach. He had deep insight into its future importance for problems of mathematical analysis and celestial mechanics and took an active part in its development. Dr. Brajraj Singh Chauhan "A Review Article on Fixed Point Theory & Its Application" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26431.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/26431/a-review-article-on-fixed-point-theory-and-its-application/dr-brajraj-singh-chauhan
Lecture #2 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
Lecture #3 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
RP 165 Solving Some Special Classes of Standard Cubic Congruence of Composite...YogeshIJTSRD
In this paper, a special class of standard cubic congruence of composite modulus is studied and formulated the solutions. The discussion is presented here. This cubic congruence has three types of solutions. In first case, it has exactly 3 incongruent solutions in second case, it has nine incongruent solutions and in third case it has twenty seven incongruent solutions. First time a suitable formulation is available for a class of standard cubic congruence. So, formulation is the merit of the paper. Prof B M Roy "RP-165: Solving Some Special Classes of Standard Cubic Congruence of Composite Modulus" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd39816.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/39816/rp165-solving-some-special-classes-of-standard-cubic-congruence-of-composite-modulus/prof-b-m-roy
Finding the Extreme Values with some Application of Derivativesijtsrd
There are many different way of mathematics rules. Among them, we express finding the extreme values for the optimization problems that changes in the particle life with the derivatives. The derivative is the exact rate at which one quantity changes with respect to another. And them, we can compute the profit and loss of a process that a company or a system. Variety of optimization problems are solved by using derivatives. There were use derivatives to find the extreme values of functions, to determine and analyze the shape of graphs and to find numerically where a function equals zero. Kyi Sint | Kay Thi Win "Finding the Extreme Values with some Application of Derivatives" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29347.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29347/finding-the-extreme-values-with-some-application-of-derivatives/kyi-sint
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The cubic diophantine equation with four unknowns given by 3 3 2 2 x y x y x y 16zw is
analyzed for its non-zero distinct integer points on it. Different patterns of integer points for the equation
under consideration are obtained. A few interesting relations between the solutions and special number
patterns namely Polygonal number, Gnomonic number, Star number and Pronic number are presented.
A Novel Approach for the Solution of a Love’s Integral Equations Using Bernst...IOSRJM
In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Love’s integral equations. The Love’s integral equation is a class of second kind Fredholm integral equations, and it can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field.This numerical technique developed by Huabsomboon et al. bases on using Taylor-series expansion [1], [2], [3] and [4]. We compare the numerical solution using Bernstein technique with the numerical solution obtained by using Chebyshev expansion. It is shown that the numerical results are excellent.
‘Six Sigma Technique’ A Journey Through its Implementationijtsrd
The manufacturing industries all over the world are facing tough challenges for growth, development and sustainability in today’s competitive environment. They have to achieve apex position by adapting with the global competitive environment by delivering goods and services at low cost, prime quality and better price to increase wealth and consumer satisfaction. Cost Management ensures profit, growth and sustainability of the business with implementation of Continuous Improvement Technique like Six Sigma. This leads to optimize Business performance. The method drives for customer satisfaction, low variation, reduction in waste and cycle time resulting into a competitive advantage over other industries which did not implement it. The main objective of this paper ‘Six Sigma Technique A Journey Through Its Implementation’ is to conceptualize the effectiveness of Six Sigma Technique through the journey of its implementation. Aditi Sunilkumar Ghosalkar "‘Six Sigma Technique’: A Journey Through its Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64546.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64546/‘six-sigma-technique’-a-journey-through-its-implementation/aditi-sunilkumar-ghosalkar
Edge Computing in Space Enhancing Data Processing and Communication for Space...ijtsrd
Edge computing, a paradigm that involves processing data closer to its source, has gained significant attention for its potential to revolutionize data processing and communication in space missions. With the increasing complexity and data volume generated by modern space missions, traditional centralized computing approaches face challenges related to latency, bandwidth, and security. Edge computing in space, involving on board processing and analysis of data, offers promising solutions to these challenges. This paper explores the concept of edge computing in space, its benefits, applications, and future prospects in enhancing space missions. Manish Verma "Edge Computing in Space: Enhancing Data Processing and Communication for Space Missions" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64541.pdf Paper Url: https://www.ijtsrd.com/computer-science/artificial-intelligence/64541/edge-computing-in-space-enhancing-data-processing-and-communication-for-space-missions/manish-verma
Dynamics of Communal Politics in 21st Century India Challenges and Prospectsijtsrd
Communal politics in India has evolved through centuries, weaving a complex tapestry shaped by historical legacies, colonial influences, and contemporary socio political transformations. This research comprehensively examines the dynamics of communal politics in 21st century India, emphasizing its historical roots, socio political dynamics, economic implications, challenges, and prospects for mitigation. The historical perspective unravels the intricate interplay of religious identities and power dynamics from ancient civilizations to the impact of colonial rule, providing insights into the evolution of communalism. The socio political dynamics section delves into the contemporary manifestations, exploring the roles of identity politics, socio economic disparities, and globalization. The economic implications section highlights how communal politics intersects with economic issues, perpetuating disparities and influencing resource allocation. Challenges posed by communal politics are scrutinized, revealing multifaceted issues ranging from social fragmentation to threats against democratic values. The prospects for mitigation present a multifaceted approach, incorporating policy interventions, community engagement, and educational initiatives. The paper conducts a comparative analysis with international examples, identifying common patterns such as identity politics and economic disparities. It also examines unique challenges, emphasizing Indias diverse religious landscape, historical legacy, and secular framework. Lessons for effective strategies are drawn from international experiences, offering insights into inclusive policies, interfaith dialogue, media regulation, and global cooperation. By scrutinizing historical epochs, contemporary dynamics, economic implications, and international comparisons, this research provides a comprehensive understanding of communal politics in India. The proposed strategies for mitigation underscore the importance of a holistic approach to foster social harmony, inclusivity, and democratic values. Rose Hossain "Dynamics of Communal Politics in 21st Century India: Challenges and Prospects" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64528.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/history/64528/dynamics-of-communal-politics-in-21st-century-india-challenges-and-prospects/rose-hossain
Assess Perspective and Knowledge of Healthcare Providers Towards Elehealth in...ijtsrd
Background and Objective Telehealth has become a well known tool for the delivery of health care in Saudi Arabia, and the perspective and knowledge of healthcare providers are influential in the implementation, adoption and advancement of the method. This systematic review was conducted to examine the current literature base regarding telehealth and the related healthcare professional perspective and knowledge in the Kingdom of Saudi Arabia. Materials and Methods This systematic review was conducted by searching 7 databases including, MEDLINE, CINHAL, Web of Science, Scopus, PubMed, PsycINFO, and ProQuest Central. Studies on healthcare practitioners telehealth knowledge and perspectives published in English in Saudi Arabia from 2000 to 2023 were included. Boland directed this comprehensive review. The researchers examined each connected study using the AXIS tool, which evaluates cross sectional systematic reviews. Narrative synthesis was used to summarise and convey the data. Results Out of 1840 search results, 10 studies were included. Positive outlook and limited knowledge among providers were seen across trials. Healthcare professionals like telehealth for its ability to improve quality, access, and delivery, save time and money, and be successful. Age, gender, occupation, and work experience also affect health workers knowledge. In Saudi Arabia, healthcare professionals face inadequate expert assistance, patient privacy, internet connection concerns, lack of training courses, lack of telehealth understanding, and high costs while performing telemedicine. Conclusions Healthcare practitioners telehealth perceptions and knowledge were examined in this systematic study. Its collection of concerned experts different personal attitudes and expertise would help enhance telehealths implementation in Saudi Arabia, develop its healthcare delivery alternative, and eliminate frequent problems. Badriah Mousa I Mulayhi | Dr. Jomin George | Judy Jenkins "Assess Perspective and Knowledge of Healthcare Providers Towards Elehealth in Saudi Arabia: A Systematic Review" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64535.pdf Paper Url: https://www.ijtsrd.com/medicine/other/64535/assess-perspective-and-knowledge-of-healthcare-providers-towards-elehealth-in-saudi-arabia-a-systematic-review/badriah-mousa-i-mulayhi
The Impact of Digital Media on the Decentralization of Power and the Erosion ...ijtsrd
The impact of digital media on the distribution of power and the weakening of traditional gatekeepers has gained considerable attention in recent years. The adoption of digital technologies and the internet has resulted in declining influence and power for traditional gatekeepers such as publishing houses and news organizations. Simultaneously, digital media has facilitated the emergence of new voices and players in the media industry. Digital medias impact on power decentralization and gatekeeper erosion is visible in several ways. One significant aspect is the democratization of information, which enables anyone with an internet connection to publish and share content globally, leading to citizen journalism and bypassing traditional gatekeepers. Another aspect is the disruption of conventional media industry business models, as traditional organizations struggle to adjust to the decrease in advertising revenue and the rise of digital platforms. Alternative business models, such as subscription models and crowdfunding, have become more prevalent, leading to the emergence of new players. Overall, the impact of digital media on the distribution of power and the weakening of traditional gatekeepers has brought about significant changes in the media landscape and the way information is shared. Further research is required to fully comprehend the implications of these changes and their impact on society. Dr. Kusum Lata "The Impact of Digital Media on the Decentralization of Power and the Erosion of Traditional Gatekeepers" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64544.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/political-science/64544/the-impact-of-digital-media-on-the-decentralization-of-power-and-the-erosion-of-traditional-gatekeepers/dr-kusum-lata
Online Voices, Offline Impact Ambedkars Ideals and Socio Political Inclusion ...ijtsrd
This research investigates the nexus between online discussions on Dr. B.R. Ambedkars ideals and their impact on social inclusion among college students in Gurugram, Haryana. Surveying 240 students from 12 government colleges, findings indicate that 65 actively engage in online discussions, with 80 demonstrating moderate to high awareness of Ambedkars ideals. Statistically significant correlations reveal that higher online engagement correlates with increased awareness p 0.05 and perceived social inclusion. Variations across colleges and a notable effect of college type on perceived social inclusion highlight the influence of contextual factors. Furthermore, the intersectional analysis underscores nuanced differences based on gender, caste, and socio economic status. Dr. Kusum Lata "Online Voices, Offline Impact: Ambedkar's Ideals and Socio-Political Inclusion - A Study of Gurugram District" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64543.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/political-science/64543/online-voices-offline-impact-ambedkars-ideals-and-sociopolitical-inclusion--a-study-of-gurugram-district/dr-kusum-lata
Problems and Challenges of Agro Entreprenurship A Studyijtsrd
Noting calls for contextualizing Agro entrepreneurs problems and challenges of the agro entrepreneurs and for greater attention to the Role of entrepreneurs in agro entrepreneurship research, we conduct a systematic literature review of extent research in agriculture entrepreneurship to overcome the study objectives of complications of agro entrepreneurs through various factors, Development of agriculture products is a key factor for the overall economic growth of agro entrepreneurs Agro Entrepreneurs produces firsthand large scale employment, utilizes the labor and natural resources, This research outlines the problems of Weather and Soil Erosions, Market price fluctuation, stimulates labor cost problems, reduces concentration of Price volatility, Dependency on Intermediaries, induces Limited Bargaining Power, and Storage and Transportation Costs. This paper mainly devoted to highlight Problems and challenges faced for the sustainable of Agro Entrepreneurs in India. Vinay Prasad B "Problems and Challenges of Agro Entreprenurship - A Study" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64540.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64540/problems-and-challenges-of-agro-entreprenurship--a-study/vinay-prasad-b
Comparative Analysis of Total Corporate Disclosure of Selected IT Companies o...ijtsrd
Disclosure is a process through which a business enterprise communicates with external parties. A corporate disclosure is communication of financial and non financial information of the activities of a business enterprise to the interested entities. Corporate disclosure is done through publishing annual reports. So corporate disclosure through annual reports plays a vital role in the life of all the companies and provides valuable information to investors. The basic objectives of corporate disclosure is to give a true and fair view of companies to the parties related either directly or indirectly like owner, government, creditors, shareholders etc. in the companies act, provisions have been made about mandatory and voluntary disclosure. The IT sector in India is rapidly growing, the trend to invest in the IT sector is rising and employment opportunities in IT sectors are also increasing. Therefore the IT sector is expected to have fair, full and adequate disclosure of all information. Unfair and incomplete disclosure may adversely affect the entire economy. A research study on disclosure practices of IT companies could play an important role in this regard. Hence, the present research study has been done to study and review comparative analysis of total corporate disclosure of selected IT companies of India and to put forward overall findings and suggestions with a view to increase disclosure score of these companies. The researcher hopes that the present research study will be helpful to all selected Companies for improving level of corporate disclosure through annual reports as well as the government, creditors, investors, all business organizations and upcoming researcher for comparative analyses of level of corporate disclosure with special reference to selected IT companies. Dr. Vaibhavi D. Thaker "Comparative Analysis of Total Corporate Disclosure of Selected IT Companies of India" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64539.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64539/comparative-analysis-of-total-corporate-disclosure-of-selected-it-companies-of-india/dr-vaibhavi-d-thaker
The Impact of Educational Background and Professional Training on Human Right...ijtsrd
This study investigated the impact of educational background and professional training on human rights awareness among secondary school teachers in the Marathwada region of Maharashtra, India. The key findings reveal that higher levels of education, particularly a master’s degree, and fields of study related to education, humanities, or social sciences are associated with greater human rights awareness among teachers. Additionally, both pre service teacher training and in service professional development programs focused on human rights education significantly enhance teacher’s knowledge, skills, and competencies in promoting human rights principles in their classrooms. Baig Ameer Bee Mirza Abdul Aziz | Dr. Syed Azaz Ali Amjad Ali "The Impact of Educational Background and Professional Training on Human Rights Awareness among Secondary School Teachers" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64529.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/education/64529/the-impact-of-educational-background-and-professional-training-on-human-rights-awareness-among-secondary-school-teachers/baig-ameer-bee-mirza-abdul-aziz
A Study on the Effective Teaching Learning Process in English Curriculum at t...ijtsrd
“One Language sets you in a corridor for life. Two languages open every door along the way” Frank Smith English as a foreign language or as a second language has been ruling in India since the period of Lord Macaulay. But the question is how much we teach or learn English properly in our culture. Is there any scope to use English as a language rather than a subject How much we learn or teach English without any interference of mother language specially in the classroom teaching learning scenario in West Bengal By considering all these issues the researcher has attempted in this article to focus on the effective teaching learning process comparing to other traditional strategies in the field of English curriculum at the secondary level to investigate whether they fulfill the present teaching learning requirements or not by examining the validity of the present curriculum of English. The purpose of this study is to focus on the effectiveness of the systematic, scientific, sequential and logical transaction of the course between the teachers and the learners in the perspective of the 5Es programme that is engage, explore, explain, extend and evaluate. Sanchali Mondal | Santinath Sarkar "A Study on the Effective Teaching Learning Process in English Curriculum at the Secondary Level of West Bengal" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd62412.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/education/62412/a-study-on-the-effective-teaching-learning-process-in-english-curriculum-at-the-secondary-level-of-west-bengal/sanchali-mondal
The Role of Mentoring and Its Influence on the Effectiveness of the Teaching ...ijtsrd
This paper reports on a study which was conducted to investigate the role of mentoring and its influence on the effectiveness of the teaching of Physics in secondary schools in the South West Region of Cameroon. The study adopted the convergent parallel mixed methods design, focusing on respondents in secondary schools in the South West Region of Cameroon. Both quantitative and qualitative data were collected, analysed separately, and the results were compared to see if the findings confirm or disconfirm each other. The quantitative analysis found that majority of the respondents 72 of Physics teachers affirmed that they had more experienced colleagues as mentors to help build their confidence, improve their teaching, and help them improve their effectiveness and efficiency in guiding learners’ achievements. Only 28 of the respondents disagreed with these statements. With majority respondents 72 agreeing with the statements, it implies that in most secondary schools, experienced Physics teachers act as mentors to build teachers’ confidence in teaching and improving students’ learning. The interview qualitative data analysis summarized how secondary school Principals use meetings with mentors and mentees to promote mentorship in the school milieu. This has helped strengthen teachers’ classroom practices in secondary schools in the South West Region of Cameroon. With the results confirming each other, the study recommends that mentoring should focus on helping teachers employ social interactions and instructional practices feedback and clarity in teaching that have direct measurable impact on students’ learning achievements. Andrew Ngeim Sumba | Frederick Ebot Ashu | Peter Agborbechem Tambi "The Role of Mentoring and Its Influence on the Effectiveness of the Teaching of Physics in Secondary Schools in the South West Region of Cameroon" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64524.pdf Paper Url: https://www.ijtsrd.com/management/management-development/64524/the-role-of-mentoring-and-its-influence-on-the-effectiveness-of-the-teaching-of-physics-in-secondary-schools-in-the-south-west-region-of-cameroon/andrew-ngeim-sumba
Design Simulation and Hardware Construction of an Arduino Microcontroller Bas...ijtsrd
This study primarily focuses on the design of a high side buck converter using an Arduino microcontroller. The converter is specifically intended for use in DC DC applications, particularly in standalone solar PV systems where the PV output voltage exceeds the load or battery voltage. To evaluate the performance of the converter, simulation experiments are conducted using Proteus Software. These simulations provide insights into the input and output voltages, currents, powers, and efficiency under different state of charge SoC conditions of a 12V,70Ah rechargeable lead acid battery. Additionally, the hardware design of the converter is implemented, and practical data is collected through operation, monitoring, and recording. By comparing the simulation results with the practical results, the efficiency and performance of the designed converter are assessed. The findings indicate that while the buck converter is suitable for practical use in standalone PV systems, its efficiency is compromised due to a lower output current. Chan Myae Aung | Dr. Ei Mon "Design Simulation and Hardware Construction of an Arduino-Microcontroller Based DC-DC High-Side Buck Converter for Standalone PV System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64518.pdf Paper Url: https://www.ijtsrd.com/engineering/mechanical-engineering/64518/design-simulation-and-hardware-construction-of-an-arduinomicrocontroller-based-dcdc-highside-buck-converter-for-standalone-pv-system/chan-myae-aung
Sustainable Energy by Paul A. Adekunte | Matthew N. O. Sadiku | Janet O. Sadikuijtsrd
Energy becomes sustainable if it meets the needs of the present without compromising the ability of future generations to meet their own needs. Some of the definitions of sustainable energy include the considerations of environmental aspects such as greenhouse gas emissions, social, and economic aspects such as energy poverty. Generally far more sustainable than fossil fuel are renewable energy sources such as wind, hydroelectric power, solar, and geothermal energy sources. Worthy of note is that some renewable energy projects, like the clearing of forests to produce biofuels, can cause severe environmental damage. The sustainability of nuclear power which is a low carbon source is highly debated because of concerns about radioactive waste, nuclear proliferation, and accidents. The switching from coal to natural gas has environmental benefits, including a lower climate impact, but could lead to delay in switching to more sustainable options. “Carbon capture and storage” can be built into power plants to remove the carbon dioxide CO2 emissions, but this technology is expensive and has rarely been implemented. Leading non renewable energy sources around the world is fossil fuels, coal, petroleum, and natural gas. Nuclear energy is usually considered another non renewable energy source, although nuclear energy itself is a renewable energy source, but the material used in nuclear power plants is not. The paper addresses the issue of sustainable energy, its attendant benefits to the future generation, and humanity in general. Paul A. Adekunte | Matthew N. O. Sadiku | Janet O. Sadiku "Sustainable Energy" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64534.pdf Paper Url: https://www.ijtsrd.com/engineering/electrical-engineering/64534/sustainable-energy/paul-a-adekunte
Concepts for Sudan Survey Act Implementations Executive Regulations and Stand...ijtsrd
This paper aims to outline the executive regulations, survey standards, and specifications required for the implementation of the Sudan Survey Act, and for regulating and organizing all surveying work activities in Sudan. The act has been discussed for more than 5 years. The Land Survey Act was initiated by the Sudan Survey Authority and all official legislations were headed by the Sudan Ministry of Justice till it was issued in 2022. The paper presents conceptual guidelines to be used for the Survey Act implementation and to regulate the survey work practice, standardizing the field surveys, processing, quality control, procedures, and the processes related to survey work carried out by the stakeholders and relevant authorities in Sudan. The conceptual guidelines are meant to improve the quality and harmonization of geospatial data and to aid decision making processes as well as geospatial information systems. The established comprehensive executive regulations will govern and regulate the implementation of the Sudan Survey Geomatics Act in all surveying and mapping practices undertaken by the Sudan Survey Authority SSA and state local survey departments for public or private sector organizations. The targeted standards and specifications include the reference frame, projection, coordinate systems, and the guidelines and specifications that must be followed in the field of survey work, processes, and mapping products. In the last few decades, there has been a growing awareness of the importance of geomatics activities and measurements on the Earths surface in space and time, together with observing and mapping the changes. In such cases, data must be captured promptly, standardized, and obtained with more accuracy and specified in much detail. The paper will also highlight the current situation in Sudan, the degree to which survey standards are used, the problems encountered, and the errors that arise from not using the standards and survey specifications. Kamal A. A. Sami "Concepts for Sudan Survey Act Implementations - Executive Regulations and Standards" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63484.pdf Paper Url: https://www.ijtsrd.com/engineering/civil-engineering/63484/concepts-for-sudan-survey-act-implementations--executive-regulations-and-standards/kamal-a-a-sami
Towards the Implementation of the Sudan Interpolated Geoid Model Khartoum Sta...ijtsrd
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Sudan is witnessing an acceleration in the processes of development and transformation in the performance of government institutions to raise the productivity and investment efficiency of the government sector. The development plans and investment opportunities have focused on achieving national goals in various sectors. This paper aims to illuminate the path to the future and provide geospatial data and information to develop the investment climate and environment for all sized businesses, and to bridge the development gap between the Sudan states. The Sudan Survey Authority SSA is the main advisor to the Sudan Government in conducting surveying, mappings, designing, and developing systems related to geospatial data and information. In recent years, SSA made a strategic partnership with the Ministry of Investment to activate Geospatial Information for Sudans Sustainable Investment and in particular, for the preparation and implementation of the Sudan investment map, based on the directives and objectives of the Ministry of Investment MI in Sudan. This paper comes within the framework of activating the efforts of the Ministry of Investment to develop technical investment services by applying techniques adopted by the Ministry and its strategic partners for advancing investment processes in the country. Kamal A. A. Sami "Activating Geospatial Information for Sudan's Sustainable Investment Map" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63482.pdf Paper Url: https://www.ijtsrd.com/engineering/information-technology/63482/activating-geospatial-information-for-sudans-sustainable-investment-map/kamal-a-a-sami
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2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30762 | Volume – 4 | Issue – 3 | March-April 2020 Page 1027
Therefore, the congruence has exactly three incongruent
solutions:
x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2.
Case-II: Let us consider that if ܽ ≠ 3݈, an even positive
integer.
Then consider: ݔ ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k =
0, 1, 2 … ….
ݔଷ
≡ ሺ2. 3୬ିଵ
k + aሻଷሺmod 4. 3୬ሻ;
≡ ሺ2. 3୬ିଵ
kሻଷ
+ 3. ሺ2. 3୬ିଵ
kሻଶ
. a + 3.2. 3୬ିଵ
k. aଶ
+
aଷ
ሺmod 4. 3୬
ሻ
≡ 4. 3୬
kሼb + 3୬ିଵ
ka + 2ଵ
. 3ଶ୬ିଷ
kሽ + aଷ
ሺmod 4. 3୬ሻ, if a is
an even positive integer
≡ aଷ
ሺmod 4. 3୬
ሻ.
Thus, x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ satisfies the said
congruence and it is a solution.
But for
k = 6, the solution becomes x ≡ 2. 3୬ିଵ
. 6 + a ሺmod 4. 3୬ሻ
≡ 4. 3୬
+ a ሺmod 4. 3୬
ሻ
≡ a ሺmod 4. 3୬
ሻ
Which is the same as for the solution for k = 0.
For k = 7, 8, … … . ., the solutions are also the same as for
k = 1, 2, …. respectively.
Thus, the congruence has exactly six incongruent solutions
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5.
Case-III: Let us now consider that ܽ = 3݈; ݈ = 1, 3, … …
Then the congruence is: ݔଷ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Also consider x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ.
ݔଷ
≡ ሺ4. 3୬ିଶ
k + 3݈ሻଷ
ሺ݉.4 ݀ 3୬
ሻ
≡ ሺ4. 3୬ିଶ
kሻଷ
+ 3. ሺ4. 3୬ିଶ
kሻଶ
. ሺ3݈ሻ + 3.4. 3୬ିଶ
k. ሺ3݈ሻଶ
+
ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Therefore, it is a solution of the congruence. But for ݇ = 9,
the solution reduces to
x ≡ 4. 3୬ିଶ
. 9 + 3݈ ሺmod 4. 3୬ሻ.
≡ 4. 3୬
+ 3݈ ሺmod 4. 3୬ሻ.
≡ 3݈ ሺmod 4. 3୬ሻ.
It is a solution of the same congruence as for ݇ = 0.
Similarly, it is also seen that ݂݇ ݎ = 10, 11, … …, the
solutions are also the same as for ݇ = 1, 2, … … ..
Therefore, it is concluded that the solutions are:
x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ;
݇ = 0, 1, 2, … . .8. These are the nine solutions.
Let us now consider that ܽ = 3݈; ݈ = 2, 4, … …
Then the congruence is: ݔଷ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Also consider x ≡ 2. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ.
ݔଷ
≡ ሺ2. 3୬ିଶ
k + 3݈ሻଷ
ሺ݉.4 ݀ 3୬
ሻ
≡ ሺ2. 3୬ିଶ
kሻଷ
+ 3. ሺ2. 3୬ିଶ
kሻଶ
. ሺ3݈ሻ + 3.2. 3୬ିଶ
k. ሺ3݈ሻଶ
+
ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ, if ܽ = 3݈, ݈ = 2, 4, … ..
Therefore, it is a solution of the congruence. But for
݇ = 18, the solution reduces to
x ≡ 2. 3୬ିଶ
. 18 + 3݈ ሺmod 4. 3୬ሻ.
≡ 4. 3୬
+ 3݈ ሺmod 4. 3୬ሻ.
≡ 3݈ ሺmod 4. 3୬ሻ.
It is a solution of the same congruence as for ݇ = 0.
Similarly, it is also seen that ݂݇ ݎ = 19, 20, … …, the
solutions are also the same as for ݇ = 1, 2, … … ..
Therefore, it is concluded that the solutions are: x ≡
2. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ;
݇ = 0, 1, 2, … . .17. These are the eighteen solutions.
Sometimes, in the cubic congruence, the integer a may not
be a perfect cube. The readers
have to make it so, by adding multiples of the modulus [1].
ILLUSTRATIONS
Example-1: Consider the congruence:
xଷ
≡ 125 ሺmod 4. 3ସሻ.
It can be written as xଷ
≡ 125 = 5ଷ
ሺmod 4. 3ସሻ with a =
5, an odd positive integer.
Such congruence has exactly three solutions which are
giving by
x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2.
≡ 4. 3ଷ
k + 5 ሺmod 4. 3ସሻ; k = 0, 1, 2.
≡ 108k + 5 ሺmod 324ሻ
≡ 5, 113, 221ሺmod 324ሻ.
Example-2: Consider the congruence xଷ
≡ 64 ሺmod 324ሻ.
It can be written as xଷ
≡ 4ଷ
ሺmod 4. 3ସሻ with a =
4, an even positive integer.
Then the six solutions are
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5 .
≡ 2. 3ଷ
k + 4 ሺmod 4. 3ସ
ሻ
≡ 54k + 4 ሺmod 324ሻ; k = 0, 1, 2, 3, 4, 5.
≡ 4, 58, 112,166, 220, 274 ሺmod 324ሻ.
Example-3: Consider the congruence
xଷ
≡ 216 ሺmod 324ሻ.
It can be written as xଷ
≡ 6ଷ
ሺmod 4. 3ସሻ with a = 6 = 3.2
Then the eighteen solutions are
x ≡ 2. 3୬ିଶ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … . . , 17 .
≡ 2. 3ଶ
k + 6 ሺmod 4. 3ସ
ሻ
≡ 18k + 6 ሺmod 324ሻ; k = 0, 1, 2, 3, … … … . , 17.
≡ 6, 24, 42, 60, 78, 96, 114,132, 150, 168, 186, 204,
222,240, 258, 276, 294, 312 ሺmod 4. 3ସ
ሻ
Example-4: Consider the congruence
xଷ
≡ 729 ሺmod 972ሻ.
It can be written as xଷ
≡ 9ଷ
ሺmod 4. 3ହሻ with a = 9 = 3.3
Then the nine solutions are
x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … . . , 8 .
≡ 4. 3ଷ
k + 9 ሺmod 4. 3ହ
ሻ
≡ 108k + 9 ሺmod 972ሻ; k = 0, 1, 2, 3, … … … . , 8.
≡ 9, 117, 225, 333, 441, 549, 657,765, 873 ሺmod 4. 3ହ
ሻ
CONCLUSION
Also, it is concluded that the congruence: xଷ
≡
aଷ
ሺmod 4. 3୬ሻ has exactly three incongruent solutions, if a
is an odd positive integer: x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ;
k = 0, 1, 2. But has exactly six incongruent solutions, if a is
an even positive integer:
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5.
If ܽ is odd multiple of three, then the congruence has nine
solutions
x ≡ 4. 3୬ିଶ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … … . .8.
3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30762 | Volume – 4 | Issue – 3 | March-April 2020 Page 1028
If ܽ is even multiple of three, then the congruence has
eighteen solutions
x ≡ 2. 3୬ିଶ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … … . .17.
MERIT OF THE PAPER
In this paper, the standard cubic congruence of composite
modulus-a multiple of four and power of three- is studied
for its solutions and is the solutions are formulated.
This lessens the labour of the readers to find the solutions.
This is the merit of the paper.
REFERENCE
[1] Roy B M, “Discrete Mathematics & Number Theory”,
1/e, Jan. 2016, Das Ganu Prakashan, Nagpur.
[2] Thomas Koshy, “Elementary Number Theory with
Applications”, 2/e (Indian print, 2009), Academic
Press.
[3] Niven I., Zuckerman H. S., Montgomery H. L. (1960,
Reprint 2008), “An Introduction to The Theory of
Numbers”, 5/e, Wiley India ( Pvt ) Ltd.
[4] Roy B M, Formulation of two special classes of
standard cubic congruence of composite modulus- a
power of Three, International Journal of Scientific
Research and Engineering Development (IJSRED),
ISSN: 2581-7175, Vol-02, Issue-03, May- June-
2019, Page-288-291.
[5] Roy B M, Solving some cubic congruence of prime
modulus, International Journal of Trend in Scientific
Research and Development (IJTSRD), ISSN: 2456-
6470, Vol-03, Issue-04, Jun-19.
[6] Roy B M, Formulation of a class of solvable standard
cubic congruence of even composite modulus,
International Journal of Advanced Research, Ideas,
and Innovations in Technology (IJARIIT), ISSN: 2454-
132X, Vol-05, Issue-01, Jan-Feb-19.
![International Journal of Trend in
Volume 4 Issue 3, April 2020
@ IJTSRD | Unique Paper ID – IJTSRD30762
Rp-72: Formulation
Cubic Congruence
Multiple
M.Sc. (Maths); Ph.D. (Hon); D.Sc. (Hon)
Jagat Arts, Commerce & I H P Science College, Goregaon (Gondia)
(Affiliated
ABSTRACT
In this study, a standard cubic congruence of composite modulus
multiple of four and power of three- is considered. After a thorough study,
formulation of the solutions of the said congruence are established in four
different cases. It is found that the standard cubic congruence under
consideration in case-I has exactly three solutions for an odd positive
integer and in case-II, has six solutions for an even positive integer. In case
III, it has nine solutions and in case-IV, the congruence has exactly eighteen
solutions. Formulations are the merit of the study.
KEYWORDS: Composite modulus, cubic residues,
Formulation
INTRODUCTION
The congruence xଷ
≡ a ሺmod mሻ is a standard cubic
congruence of prime or composite
congruence are said to be solvable if ܽ is a cubic residue of
m. In this study, the author considered the congruence:
xଷ
≡ aଷ
ሺmod 4. 3୬ሻ for his study and the formulation of
its solutions. It is always solvable. But nothing is said to
test of the solvability of the said congruence in the
literature. It seems that no earlier mathematicians were
serious about the said congruence except two: Koshy and
Zuckerman.
Koshy only defined a cubic residue [1] while Zuckerman
had just started to find the solutions [2] but stopped
abruptly without mentioning any method of solutions.
Finding no formulation of the solutions of the congruence,
the author started his study and formulation
solutions of cubic congruence Of composite modulus.
Many such congruence are formulated and the papers
have been published in different International Journals
[4], [5], [6]. Here in this paper, the author considered the
cubic congruence of the type: xଷ
≡ aଷ
ሺmod
International Journal of Trend in Scientific Research and Development (IJTSRD)
2020 Available Online: www.ijtsrd.com e-ISSN: 2456
30762 | Volume – 4 | Issue – 3 | March-April
72: Formulation of Solutions of a Standard
Congruence of Composite Modulus
Multiple of Four and Power of Three
Prof B M Roy
M.Sc. (Maths); Ph.D. (Hon); D.Sc. (Hon), Head, Department of Mathematics
Jagat Arts, Commerce & I H P Science College, Goregaon (Gondia), Maharashtra, India
(Affiliated to R T M Nagpur University)
In this study, a standard cubic congruence of composite modulus --a
considered. After a thorough study,
formulation of the solutions of the said congruence are established in four
different cases. It is found that the standard cubic congruence under
I has exactly three solutions for an odd positive
II, has six solutions for an even positive integer. In case-
IV, the congruence has exactly eighteen
solutions. Formulations are the merit of the study.
Composite modulus, cubic residues, Standard cubic Congruence,
How to cite this paper
"Rp-72: Formulation of Solutions of a
Standard Cubic Congruence o
Composite Modulus
and Power of
Three" Published in
International
Journal of Trend in
Scientific Research
and Development
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is a standard cubic
modulus. Such
is a cubic residue of
m. In this study, the author considered the congruence:
the formulation of
But nothing is said to
test of the solvability of the said congruence in the
ure. It seems that no earlier mathematicians were
serious about the said congruence except two: Koshy and
Koshy only defined a cubic residue [1] while Zuckerman
solutions [2] but stopped
any method of solutions.
Finding no formulation of the solutions of the congruence,
the author started his study and formulation of the
solutions of cubic congruence Of composite modulus.
Many such congruence are formulated and the papers
hed in different International Journals
[4], [5], [6]. Here in this paper, the author considered the
ሺmod 4. 3୬ሻ.
PROBLEM STATEMENT
Here, the problem under consideration is to find a
formulation of the solutions of
of composite modulus of the type:
xଷ
≡ aଷ
ሺmod 4. 3୬ሻ in four different cases:
Case-I: If ܽ ് 3݈, an odd positive integer;
Case-II: If ܽ ് 3݈, an even positive integer;
Case-III: If ܽ ൌ 3݈, an odd positive integer;
Case-IV: : If ܽ ൌ 3݈, an even positive integer.
ANALYSIS AND RESULT
Case-I: Let ܽ ് 3݈ and odd positive integer.
Now consider the congruence
Then, for the solutions, consider
a ሺmod 4. 3୬ሻ; k ൌ 0, 1, 2 … ….
Therefore, xଷ
≡ ሺ4. 3୬ିଵ
k a
≡ ሺ4. 3୬ିଵ
kሻଷ
3. ሺ4. 3୬ିଵ
kሻ
aଷ
ሺmod 4. 3୬
ሻ
≡ 4. 3୬
kሼaଶ
4. 3୬ିଵ
ka 4ଶ
ܽ ് 3݉ is odd;
≡ aଷ
ሺmod 4. 3୬
ሻ.
Thus, it is a solution of the congruence. But for
݇ ൌ 3,4,5 … . ,, it is seen that the solution is the same as for
݇ ൌ 0,1, 2.
Scientific Research and Development (IJTSRD)
ISSN: 2456 – 6470
April 2020 Page 1026
a Standard
f Composite Modulus-a
f Three
f Mathematics,
Maharashtra, India
How to cite this paper: Prof B M Roy
72: Formulation of Solutions of a
Standard Cubic Congruence of
Composite Modulus-a Multiple of Four
and Power of
Three" Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
3, April 2020, pp.1026-1028, URL:
www.ijtsrd.com/papers/ijtsrd30762.pdf
Copyright © 2020 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
Here, the problem under consideration is to find a
formulation of the solutions of Standard cubic congruence
of composite modulus of the type:
in four different cases:
an odd positive integer;
an even positive integer;
an odd positive integer;
an even positive integer.
and odd positive integer.
Now consider the congruence xଷ
≡ aଷ
ሺmod 4. 3୬ሻ.
Then, for the solutions, consider x ≡ 4. 3୬ିଵ
k
.
a ሻଷ
ሺmod 4. 3୬ሻ
ሻଶ
. a 3.4. 3୬ିଵ
k. aଶ
ଶ
. 3ଶ୬ିଷ
kሽ aଷ
ሺmod 4. 3୬
ሻ, if
Thus, it is a solution of the congruence. But for
it is seen that the solution is the same as for
IJTSRD30762](https://image.slidesharecdn.com/207rp-72formulationofsolutionsofastandardcubiccongruenceofcompositemodulus-amultipleoffourand-200608072635/85/Rp-72-Formulation-of-Solutions-of-a-Standard-Cubic-Congruence-of-Composite-Modulus-a-Multiple-of-Four-and-Power-of-Three-1-320.jpg)
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Therefore, the congruence has exactly three incongruent
solutions:
x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2.
Case-II: Let us consider that if ܽ ≠ 3݈, an even positive
integer.
Then consider: ݔ ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k =
0, 1, 2 … ….
ݔଷ
≡ ሺ2. 3୬ିଵ
k + aሻଷሺmod 4. 3୬ሻ;
≡ ሺ2. 3୬ିଵ
kሻଷ
+ 3. ሺ2. 3୬ିଵ
kሻଶ
. a + 3.2. 3୬ିଵ
k. aଶ
+
aଷ
ሺmod 4. 3୬
ሻ
≡ 4. 3୬
kሼb + 3୬ିଵ
ka + 2ଵ
. 3ଶ୬ିଷ
kሽ + aଷ
ሺmod 4. 3୬ሻ, if a is
an even positive integer
≡ aଷ
ሺmod 4. 3୬
ሻ.
Thus, x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ satisfies the said
congruence and it is a solution.
But for
k = 6, the solution becomes x ≡ 2. 3୬ିଵ
. 6 + a ሺmod 4. 3୬ሻ
≡ 4. 3୬
+ a ሺmod 4. 3୬
ሻ
≡ a ሺmod 4. 3୬
ሻ
Which is the same as for the solution for k = 0.
For k = 7, 8, … … . ., the solutions are also the same as for
k = 1, 2, …. respectively.
Thus, the congruence has exactly six incongruent solutions
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5.
Case-III: Let us now consider that ܽ = 3݈; ݈ = 1, 3, … …
Then the congruence is: ݔଷ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Also consider x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ.
ݔଷ
≡ ሺ4. 3୬ିଶ
k + 3݈ሻଷ
ሺ݉.4 ݀ 3୬
ሻ
≡ ሺ4. 3୬ିଶ
kሻଷ
+ 3. ሺ4. 3୬ିଶ
kሻଶ
. ሺ3݈ሻ + 3.4. 3୬ିଶ
k. ሺ3݈ሻଶ
+
ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Therefore, it is a solution of the congruence. But for ݇ = 9,
the solution reduces to
x ≡ 4. 3୬ିଶ
. 9 + 3݈ ሺmod 4. 3୬ሻ.
≡ 4. 3୬
+ 3݈ ሺmod 4. 3୬ሻ.
≡ 3݈ ሺmod 4. 3୬ሻ.
It is a solution of the same congruence as for ݇ = 0.
Similarly, it is also seen that ݂݇ ݎ = 10, 11, … …, the
solutions are also the same as for ݇ = 1, 2, … … ..
Therefore, it is concluded that the solutions are:
x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ;
݇ = 0, 1, 2, … . .8. These are the nine solutions.
Let us now consider that ܽ = 3݈; ݈ = 2, 4, … …
Then the congruence is: ݔଷ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
Also consider x ≡ 2. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ.
ݔଷ
≡ ሺ2. 3୬ିଶ
k + 3݈ሻଷ
ሺ݉.4 ݀ 3୬
ሻ
≡ ሺ2. 3୬ିଶ
kሻଷ
+ 3. ሺ2. 3୬ିଶ
kሻଶ
. ሺ3݈ሻ + 3.2. 3୬ିଶ
k. ሺ3݈ሻଶ
+
ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ
≡ ሺ3݈ሻଷ
ሺmod 4. 3୬
ሻ, if ܽ = 3݈, ݈ = 2, 4, … ..
Therefore, it is a solution of the congruence. But for
݇ = 18, the solution reduces to
x ≡ 2. 3୬ିଶ
. 18 + 3݈ ሺmod 4. 3୬ሻ.
≡ 4. 3୬
+ 3݈ ሺmod 4. 3୬ሻ.
≡ 3݈ ሺmod 4. 3୬ሻ.
It is a solution of the same congruence as for ݇ = 0.
Similarly, it is also seen that ݂݇ ݎ = 19, 20, … …, the
solutions are also the same as for ݇ = 1, 2, … … ..
Therefore, it is concluded that the solutions are: x ≡
2. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ;
݇ = 0, 1, 2, … . .17. These are the eighteen solutions.
Sometimes, in the cubic congruence, the integer a may not
be a perfect cube. The readers
have to make it so, by adding multiples of the modulus [1].
ILLUSTRATIONS
Example-1: Consider the congruence:
xଷ
≡ 125 ሺmod 4. 3ସሻ.
It can be written as xଷ
≡ 125 = 5ଷ
ሺmod 4. 3ସሻ with a =
5, an odd positive integer.
Such congruence has exactly three solutions which are
giving by
x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2.
≡ 4. 3ଷ
k + 5 ሺmod 4. 3ସሻ; k = 0, 1, 2.
≡ 108k + 5 ሺmod 324ሻ
≡ 5, 113, 221ሺmod 324ሻ.
Example-2: Consider the congruence xଷ
≡ 64 ሺmod 324ሻ.
It can be written as xଷ
≡ 4ଷ
ሺmod 4. 3ସሻ with a =
4, an even positive integer.
Then the six solutions are
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5 .
≡ 2. 3ଷ
k + 4 ሺmod 4. 3ସ
ሻ
≡ 54k + 4 ሺmod 324ሻ; k = 0, 1, 2, 3, 4, 5.
≡ 4, 58, 112,166, 220, 274 ሺmod 324ሻ.
Example-3: Consider the congruence
xଷ
≡ 216 ሺmod 324ሻ.
It can be written as xଷ
≡ 6ଷ
ሺmod 4. 3ସሻ with a = 6 = 3.2
Then the eighteen solutions are
x ≡ 2. 3୬ିଶ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … . . , 17 .
≡ 2. 3ଶ
k + 6 ሺmod 4. 3ସ
ሻ
≡ 18k + 6 ሺmod 324ሻ; k = 0, 1, 2, 3, … … … . , 17.
≡ 6, 24, 42, 60, 78, 96, 114,132, 150, 168, 186, 204,
222,240, 258, 276, 294, 312 ሺmod 4. 3ସ
ሻ
Example-4: Consider the congruence
xଷ
≡ 729 ሺmod 972ሻ.
It can be written as xଷ
≡ 9ଷ
ሺmod 4. 3ହሻ with a = 9 = 3.3
Then the nine solutions are
x ≡ 4. 3୬ିଶ
k + 3݈ ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … . . , 8 .
≡ 4. 3ଷ
k + 9 ሺmod 4. 3ହ
ሻ
≡ 108k + 9 ሺmod 972ሻ; k = 0, 1, 2, 3, … … … . , 8.
≡ 9, 117, 225, 333, 441, 549, 657,765, 873 ሺmod 4. 3ହ
ሻ
CONCLUSION
Also, it is concluded that the congruence: xଷ
≡
aଷ
ሺmod 4. 3୬ሻ has exactly three incongruent solutions, if a
is an odd positive integer: x ≡ 4. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ;
k = 0, 1, 2. But has exactly six incongruent solutions, if a is
an even positive integer:
x ≡ 2. 3୬ିଵ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, 4, 5.
If ܽ is odd multiple of three, then the congruence has nine
solutions
x ≡ 4. 3୬ିଶ
k + a ሺmod 4. 3୬ሻ; k = 0, 1, 2, 3, … … . .8.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)