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Numbers and Its Types in Mathematics

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What are numbers in Math ?
Real Numbers
Rational Numbers
Irrational Number
Operation on Real Numbers
What are Integers?
Even Numbers
Odd Numbers
Whole Numbers
Prime Numbers
Co-Prime Numbers
Composite Number
Prime Number vs. Composite Numbers
What is a Perfect Number?
Complex Numbers
 Fractions
Improper Fraction
Proper Fraction
 Mixed Fractions

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Numbers and Its Types in Mathematics

  1. 1. Numbers And Their Types in Mathematics Lecture Slides By Adil Aslam Email Me : adilaslam5959@gmail.com
  2. 2. What are numbers in Math ? • A number is a mathematical tool which is used in counting individual quantities, calculating and quantifying. • In general, decimal number system is used which consists of 10 digits, from 0 to 9. • A Binary Number is made up of only 0s and 1s. • Example of decimal number is 7. • Example of Binary number is 1001 Lecture Slides By Adil Aslam 2
  3. 3. Various Operations Performed on Numbers • Addition: It is the process of finding out single number or fraction equal to two or more quantities taken together. • For Example 2+1 = 3 • Subtraction: It is the process of finding out the quantity left when a smaller quantity (number/ fraction) is reduced from a larger one. • For Example 4-2 = 2 Lecture Slides By Adil Aslam 3
  4. 4. Various Operations Performed on Numbers • Multiplication: It signifies repeated addition. If a number has to be repeatedly added then that number is multiplicand. The number of multiplicands considered for addition is multiplier. The sum of the repetition is the product. • For Example 4*2 = 8 • Division: It is a reversal of multiplication. In this we find how often a given number called divisor is contained in another given number called dividend. The number expressing this is called the quotient and the excess of the dividend over the product of the divisor and quotient is called remainder. • For Example 6/3 = 2 Lecture Slides By Adil Aslam 4
  5. 5. Lecture Slides By Adil Aslam 5 Naturals 1, 2, 3... Wholes 0 Integers -3 -19 Rational 2 3 6 1 4 -2.65 Irrational  2 Real Numbers Imaginary Numbers 1 Complex Numbers
  6. 6. Real Numbers • Real Numbers include all the rational and irrational numbers. • Real numbers are denoted by the letter R. • Real numbers consist of the natural numbers, whole numbers, integers, rational, and irrational numbers. • The decimal expansion of a real number is either terminating, repeating or non-terminating and non- repeating. Lecture Slides By Adil Aslam 6
  7. 7. Real Numbers This Venn Diagram displays the Sets of Real Number (Rational, Irrational, Integers, Wholes, and Naturals) Lecture Slides By Adil Aslam 7 Naturals 1, 2, 3... Wholes 0 Integers -3 -19 Rational 2 3 6 1 4 -2.65 Irrational  2 Real Numbers 5 5 0
  8. 8. Rational Numbers • A rational number is part of a whole expressed as a fraction, decimal or a percentage. • A number is rational if we can write it as a fraction where the top number of the fraction and bottom number are both whole numbers. • The term rational is derived from the word 'ratio' because the rational numbers are figures which can be written in the ratio form. • Every whole number, including negative numbers and zero, is a rational number. This is because every whole number ‘n’ can be written in the form n/1. Lecture Slides By Adil Aslam 8
  9. 9. Rational Numbers • Examples of Rational Numbers • The number 8 is a rational number because it can be written as the fraction 8/1. • Likewise, 3/4 is a rational number because it can be written as a fraction. • 1.5 is rational, because it can be written as the ratio 3/2. • 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3. • Recurring decimals such as 0.26262626…, all integers and all finite decimals, such as 0.241, are also rational numbers. Lecture Slides By Adil Aslam 9
  10. 10. Irrational Number • An irrational number is any number that is not rational. It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction). • An irrational number has endless non-repeating digits to the right of the decimal point. Lecture Slides By Adil Aslam 10
  11. 11. Irrational Number • Examples of irrational number • For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers. • π = 3.141592…is an irrational number because it cannot be expressed as a fraction of two whole numbers and it has no accurate decimal equivalent. • Euler's Number (e) is another famous irrational number. Like Pi, Euler's Number has been calculated to many decimal places without any pattern showing. The value of e is 2.7182818284590452353… and keeps going much like the value of Pi. Lecture Slides By Adil Aslam 11
  12. 12. Operation on Real Numbers • Operations on rational and irrational numbers: • The sum of a rational and an irrational number is irrational. For example: 3 + √3 = 4.73205…., which is irrational since it does not follow any specific pattern and goes on endlessly after the decimal. • The difference of a rational and an irrational number is irrational. For example: 3 - √3 = 2.73205…., which is irrational since it does not follow any specific pattern and goes on endlessly after the decimal. Lecture Slides By Adil Aslam 12
  13. 13. Operation on Real Numbers • Operations on rational and irrational numbers: • The product of a non-zero rational with an irrational number is irrational. For example: 3 × √3 = 5.19615…. which is irrational. • The quotient of a non-zero rational with an irrational number is irrational. For example: 3 ÷ √3 = 1.73205…. which is irrational. Lecture Slides By Adil Aslam 13
  14. 14. What are Integers? • Integers are a type of rational numbers. Rational numbers of the form p/q where q= ±1 are called integers. They are denoted by “Z”. • Integers are of two types : • Negative integers Negative integers are the set of negative numbers before 0. They do not have any fractional or decimal part. E.g. -1, -2, -3 and so on. • Positive integers / Whole numbers Positive integers / whole numbers are the set of natural numbers including zero. They do not have any fractional or decimal part. • To distinguish between the positive and negative sides of 0, opposite signs are used, i.e. positive (+) and negative (-). Lecture Slides By Adil Aslam 14
  15. 15. What are Integers? Lecture Slides By Adil Aslam 15
  16. 16. Integers • Additive inverse: • The opposite of an integer is called its negative or additive inverse. • 3 + (-3) = 0. So 3 is the additive inverse of -3 and vice versa. • Each integer on the number line, except 0, consists of its mirror image of the opposite sign. E.g. Mirror image of 6 will be -6,that of -1 will be 1 etc. • Zero is lesser than every positive integer and greater than every negative integer. • It is not possible to find the greatest positive or least negative number. Lecture Slides By Adil Aslam 16
  17. 17. Integers • Absolute value of an integer: • The absolute value of an integer is the numerical value without taking into account its sign. The modulus sign represents the absolute value of the integer. • E.g.|-3|=3, |5|=5. Thus, the absolute value of an integer is either greater than or equal to the integer. Lecture Slides By Adil Aslam 17
  18. 18. Integers Integer Operations • Addition + • Subtraction - • Multiplication x • Division ÷ Lecture Slides By Adil Aslam 18
  19. 19. Integers Rules for operations on integers: • Addition: • Addition of 2 positive or 2 negative integers: • Add their absolute values and prefix the sign of addends to the sum. • -2 + (-3) = -5 • 3 + 4 = 7 • Addition of 1 positive and 1 negative integer: • Find the difference of their absolute values and prefix the sign of the integer whose absolute value is greater. • 2 + (-3) = -(3-2) = -1 • (-4) + 1 = -3 Lecture Slides By Adil Aslam 19
  20. 20. Your Task! (Addition) 1) 5 + 6 = 2) -3 + (-2) = 3) -6 + 5 = 4) 8 + (-7) = 5) -9 + 9 = Lecture Slides By Adil Aslam 20
  21. 21. Your Task! (Addition) 1) 5 + 6 = 11 2) -3 + (-2) = -5 3) -6 + 5 = -1 4) 8 + (-7) = 1 5) -9 + 9 = 0 Lecture Slides By Adil Aslam 21
  22. 22. Integers Rules for operations on integers: • Subtraction: • Subtracting 1 negative from another negative: • When there are 2 negative(-ve) signs placed side by side with no numeral in between, the 2 like signs become a plus sign. • -2 – (-3) = -2 + 3 = 1 • Subtraction of 1 negative from 1 positive: • Convert the like signs in case of negative signs and then add the integers. • 3 - (-2) = 3 + 2 = 5 • Subtraction of 1 positive from 1 negative: • Convert the signs and then add the absolute values of integers add prefix minus sign. • -3 – (+2) = -3 -2 = -5 Lecture Slides By Adil Aslam 22
  23. 23. Your Task! (Subtraction) 1) 5 – 2 = 2) -3 – 4 = 3) -1 – (-2) = 4) -5 – (-3) = 5) 7 – (-6) = Lecture Slides By Adil Aslam 23
  24. 24. Your Task! (Subtraction) 1) 5 – 2 = 3 2) -3 – 4 = -7 3) -1 – (-2) = 1 4) -5 – (-3) = -2 5) 7 – (-6) = 13 Lecture Slides By Adil Aslam 24
  25. 25. Integers Rules for operations on integers: • Multiplication: • Multiplication of 2 positive integers: • Multiply the values and prefix plus sign to the product. 2 x 3 = +6 or 6 • Multiplication of a positive and a negative integer: • Multiply their absolute values and prefix minus sign to the product. • 2 x (-3) = -6 • Multiplication of 2 negative integers: • Multiply the absolute values and prefix plus sign to the product. • (-2) x (-3) = +6 or 6 Lecture Slides By Adil Aslam 25
  26. 26. Your Task! (Multiplication) 1) 6 x (-3) = 2) 3 x 3 = 3) -4 x 5 = 4) -6 x (-2) = 5) -7 x (-8) = Lecture Slides By Adil Aslam 26
  27. 27. Your Task! (Multiplication) 1) 6 x (-3) = -18 2) 3 x 3 = 9 3) -4 x 5 = -20 4) -6 x (-2) = 12 5) -7 x (-8) = 56 Lecture Slides By Adil Aslam 27
  28. 28. Integers Rules for operations on integers: • Division: • Division of integers with like signs: • Divide their absolute values and prefix plus sign. • (-3) ÷ (-2) = +3/2 or 3/2 • Division of integers with like signs: • Divide their absolute values and prefix minus sign. • (3) ÷ (-2) = -3/2 Lecture Slides By Adil Aslam 28
  29. 29. Your Task! (Division) 1) 18 ÷ (-2) = 2) -48 ÷ (-6) = 3) -27 ÷ 9 = 4) 64 ÷ 8 = 5) 30 ÷ (-5) = Lecture Slides By Adil Aslam 29
  30. 30. Your Task! (Division) 1) 18 ÷ (-2) = -9 2) -48 ÷ (-6) = 8 3) -27 ÷ 9 = -3 4) 64 ÷ 8 = 8 5) 30 ÷ (-5) = -6 Lecture Slides By Adil Aslam 30
  31. 31. Natural Numbers • This group of numbers starts at 1. • It includes 1, 2, 3, 4, 5, and so on. • Zero is not in this group. • This group has no negative numbers. • There are no numbers with decimals in this group. • Natural numbers are the set of positive integers, that is, integers from 1 to ∞ excluding fractional n decimal part. • Natural numbers could be seen as: • 1, 2, 3, 4, 5, .... Lecture Slides By Adil Aslam 31
  32. 32. What is Zero ? • Zero is the only whole number which is not a natural number. It is represented by 0. "Zero is the only difference between natural and whole numbers.“ • Properties of ZERO (0) • Zero added to any number gives the number itself. Hence, 0 is the additive identity for all numbers. • Multiplication with zero- 0 multiplied with any number always gives 0. • Zero divided my any number(except 0) gives 0. • Any number divided by 0 is not defined. • Zero raised to any power(except 0) gives 0. • Zero to the power 0 (00) is not defined. Lecture Slides By Adil Aslam 32
  33. 33. Natural Numbers • Natural Numbers can further be sub-divided into different types :- 1) Even and odd Numbers 2) Prime Numbers 3) Relatively Prime/ co-prime numbers 4) Composite Numbers 5) Perfect Numbers Lecture Slides By Adil Aslam 33
  34. 34. Even Numbers • An even number is any number that can be divided by 2. • For example, 12 can be divided by 2, so 12 is even. • We saw in divisibility rules that a number is divisible by 2 if its last digit is 0,2,4,6,or 8. • Therefore, any number whose last digit is 0, 2, 4, 6, or 8 is an even number • Other examples of even numbers are 58, 44884, 998632, 98, 48, and 10000000 Lecture Slides By Adil Aslam 34
  35. 35. Even Numbers • Formal definition of an even number: • A number n is even if there exist a number k, such that n = 2k where k is an integer. • This is formal way of saying that if n is divided by 2, we always get a quotient k with no remainder . • Having no remainder means that n can in fact be divided by 2 Lecture Slides By Adil Aslam 35
  36. 36. Odd Numbers • An odd number is any number that cannot be divided by 2. • For example, 25 cannot be divided by 2, so 25 is odd. • We saw in divisibility rules that a number is divisible by 2 if its last digit is 0,2,4,6,or 8. • Therefore, any number whose last digit is not 0, 2, 4, 6, or 8 is an odd number • Other examples of odd numbers are 53, 881, 238637, 99, 45, and 100000023 Lecture Slides By Adil Aslam 36
  37. 37. Odd Numbers • Formal definition of an odd number: • A number n is odd if there exist a number k, such that n = 2k + 1 where k is an integer. • This is formal way of saying that if n is divided by 2, we always get a quotient k with a remainder of 1. • Having a remainder of 1 means that n cannot in fact be divided by 2 Lecture Slides By Adil Aslam 37
  38. 38. Even and Odd Numbers Basic operations with even and odd numbers Lecture Slides By Adil Aslam 38 Addition Rule Example even + even = even 4 + 2 = 6 even + odd = odd 6 + 3 = 9 odd + odd = even 13 + 13 = 26
  39. 39. Even and Odd Numbers Basic operations with even and odd numbers Lecture Slides By Adil Aslam 39 Multiplication Rule Example even × even = even 2 × 6 = 12 even × odd = even 8 × 3 = 24 odd × odd = odd 3 × 5 = 15
  40. 40. Even and Odd Numbers Basic operations with even and odd numbers Lecture Slides By Adil Aslam 40 Subtraction Rule Example even − even = even 8 − 4 = 4 even − odd = odd 8 − 5 = 3 odd − odd = even 9 − 5 = 4
  41. 41. Whole Numbers • This group has all of the Natural Numbers in it plus the number 0. • They do not have any decimal or fractional part. • Whole numbers could be seen as: • 0, 1, 2, 3, 4, 5, ... Lecture Slides By Adil Aslam 41
  42. 42. Closure Property Of Whole Numbers • Addition of whole numbers always gives a whole number. Thus whole numbers are closed under addition. • For Example, 3+8 = 11 • Subtraction of whole numbers does not always give a whole number. Thus whole numbers are not closed under subtraction. • For Example, 5-7 = -2, which is a negative integer and not a whole number. • For Example, 9-6 = 3 which is whole number. • Lecture Slides By Adil Aslam 42
  43. 43. Closure Property Of Whole Numbers • Multiplication of whole numbers always gives a whole number. Thus whole numbers are closed under multiplication. • For Example, 5x8 = 40 • Dividing a whole number by another does not always give a whole number. Thus whole numbers are not closed under division. • For Example, 4÷7 = a fraction, thus it will not be a whole number. Lecture Slides By Adil Aslam 43
  44. 44. Prime Numbers • A Prime Number can be divided evenly only by 1 or itself. • And it must be a whole number greater than 1. • For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. If we list the factors of 29, we only have 1 and 29. That's two factors. So we say that 29 is a prime number, but 28 isn't. • Note that the definition of a prime number doesn't allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have exactly two factors, not "at most two" or anything like that. When a number has more than two factors it is called a composite number. Lecture Slides By Adil Aslam 44
  45. 45. Prime Numbers • Some facts: • The only even prime number is 2. All other even numbers can be divided by 2. • If the sum of a number's digits is a multiple of 3, that number can be divided by 3. • No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5. • Zero and 1 are not considered prime numbers. • Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime. Lecture Slides By Adil Aslam 45
  46. 46. Prime Numbers • All prime numbers less than 100 Lecture Slides By Adil Aslam 46
  47. 47. Co-Prime Numbers • A set of numbers which do not have any other common factor other than 1 are called co-prime or relatively prime numbers. • This means those numbers whose HCF is 1. • For example, 8 and 9 have no other common factor other than 1 so they are co-prime numbers. Lecture Slides By Adil Aslam 47
  48. 48. Co-Prime Numbers Examples • Example 1: 21 and 22 • 21 and 22: • The factors of 21 are 1, 3, 7 and 21. • The factors of 22 are 1, 2, 11 and 22. • Here 21 and 22 have only one common factor that is 1.Hence their HCF is 1 and are co-prime. • Examples 2 : 21 and 27 • 21 and 27: • The factors of 21 are 1, 3, 7 and 21. • The factors of 27 are 1, 2, 3, 9 and 27. • Here 21 and 27 have two common factors they are 1 and 3. HCF is 3 and they are not co-prime. Lecture Slides By Adil Aslam 48
  49. 49. Co-Prime Numbers Properties of co-prime numbers • 1 is co-prime with every number. • Every prime number is co-prime to each other: • As every prime number have only two factors 1 and the number itself, the only common factor of two prime numbers will be 1.For example, 2 and 3 are two prime numbers. Factors of 2 are 1, 2 and factors of 3 are 1, 3. The only common factor is 1 and hence is co-prime. • Any two successive numbers/ integers are always co- prime: • Take any consecutive number such as 2, 3 or 3, 4 or 5, 6 and so on; they have 1 as their HCF. • The sum of any two co-prime numbers are always co- prime with their product: • 2 and 3 are co-prime and have 5 as their sum (2+3) and 6 as the product (2×3). Hence, 5 and 6 are co-prime to each other. Lecture Slides By Adil Aslam 49
  50. 50. Composite Number • A composite number is a positive integer that has a factor other than 1 and itself. For instance, 6 is a composite number because: • it is positive • it has a factor other than 1 and itself (both 2 and 3 are factors of 6) • A composite number has more than two factors. • The number 1 is neither prime nor composite. Lecture Slides By Adil Aslam 50
  51. 51. Composite Number Examples of Composite Numbers • 4 • 4 is composite because 2 is a factor • 8 • 8 is a composite number because it has 4 and 2 as factors • 9 • 9 is a composite number because the factors of 9 are 1 x 9, 3 x 3 Lecture Slides By Adil Aslam 51
  52. 52. Composite Number Examples of not Composite Numbers • 5 • 5 is a not a composite number because its only factors are 1 and itself (5). 5 is, however, a prime number. • 2 • 2 is a not a composite number because its only factors are 1 and itself (2). 2 is, however, a prime number . 2 is also the even number that is not composite. Every other even number is indeed a composite number. • 7 • 7 is a not a composite number because its only factors are 1 and itself (7). 7 is, however, a prime number. Lecture Slides By Adil Aslam 52
  53. 53. Prime Number vs. Composite Numbers Lecture Slides By Adil Aslam 53
  54. 54. What is a Perfect Number? • A perfect number is a whole number, an integer greater than zero; and when you add up all of the factors less than that number, you get that number. • Examples: • The factors of 6 are 1, 2, 3 and 6. 1 + 2 + 3 = 6 • The factors of 28 are 1, 2, 4, 7, 14 and 28. 1 + 2 + 4 + 7 + 14 = 28 • The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496. 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 • The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128. I'll let you add them up. Lecture Slides By Adil Aslam 54
  55. 55. What is a Perfect Number? • How many perfect numbers are known? • So far, according to the Mersenne organization, there are 37 known Mersenne prime numbers. This means that there are 37 known "perfect" numbers. The newest prime number was discovered on January 27, 1998. • Are there any odd perfect numbers? • Nobody has found any odd perfect numbers, but we do not know if any odd ones exist. Lecture Slides By Adil Aslam 55
  56. 56. Perfect Number Lecture Slides By Adil Aslam 56 These number are Perfect
  57. 57. Complex Numbers • Complex numbers are the numbers which have both real and imaginary parts. • The imaginary part of the complex number can’t be represented on number line. • They are of the form a+ bi where a and b are real numbers while i is an imaginary part (i = √1), “a” represents the real part while "bi" represents the imaginary part. • Complex were introduced to find the solution of equation X2 + 1 = 0 Here x=√-1, that is represented as i. Lecture Slides By Adil Aslam 57
  58. 58. Complex Numbers • A number such as 3i is a purely imaginary number • A number such as 6 is a purely real number • 6 + 3i is a complex number • x + iy is the general form of a complex number • If x + iy = 6 – 4i then x = 6 and y = – 4 • The ‘real part’ of 6 – 4i is 6 Lecture Slides By Adil Aslam 58
  59. 59. Complex Numbers • The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. • The first four powers of i establish an important pattern and should be memorized. Lecture Slides By Adil Aslam 59 Powers of i 1 2 3 4 1 1i i i i i i     
  60. 60. Complex Numbers • By definition • Consider powers if i Lecture Slides By Adil Aslam 60 2 1 1i i     2 3 2 4 2 2 5 4 1 1 1 1 1 ... i i i i i i i i i i i i i                 
  61. 61. Lecture Slides By Adil Aslam 61 Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals (no fractions) pi, e Imaginary i, 2i, -3-7i, etc.
  62. 62. Fractions • The word fraction means a part of something that is whole i.e. its value should be less than 1. • Fractions are a type of rational numbers (which are of the form p/q, where p and q are integers and q≠0), and have numerator less than the denominator (p < q) and both p and q are in the lowest terms. • A fraction is in the lowest terms if the only common factor between the numerator and denominator is 1. • E.g. 5/10 is not in the lowest terms since the common factors are 1 and 5; whereas 3/7 is in its lowest form. • A fraction can be treated as a division problem. Lecture Slides By Adil Aslam 62
  63. 63. Fractions • A fraction is formed by dividing a whole into a number of parts Lecture Slides By Adil Aslam 63 3 2 I’m the NUMERATOR. I tell you the number of parts I’m the DENOMINATOR. I tell you the name of part
  64. 64. ¾ looks like • ¾ Looks like this: Lecture Slides By Adil Aslam 64 1/43 4 1/4 1/4
  65. 65. ¾ looks like • ¾ Looks like this: 1/4 3 4 1/4 1/4 Lecture Slides By Adil Aslam 65
  66. 66. What fraction of the balls are green? Lecture Slides By Adil Aslam 66 1 4
  67. 67. What fraction of the balls are green? Lecture Slides By Adil Aslam 67 1 6
  68. 68. What fraction of the balls are green? 4 6 Lecture Slides By Adil Aslam 68
  69. 69. Proper Fraction • Fractions in which the numerator is less than the denominator are known as proper fractions. • Examples : 3/4, 5/10, 3/9 etc. Lecture Slides By Adil Aslam 69
  70. 70. Improper Fraction • Fractions in which the denominator is less than the numerator are known as improper fractions. • Examples : 5/4, 9/8, 11/2 etc. Lecture Slides By Adil Aslam 70
  71. 71. Mixed Fractions • Fractions which are a combination of a whole number and a fraction are called mixed fractions or mixed numbers. • Examples : 11/4, 31/9 etc. • Mixed fractions are an example of improper fractions. • Improper fractions can be converted into mixed fractions and vice versa. Lecture Slides By Adil Aslam 71
  72. 72. Like and Unlike Fractions • Like fractions are those which have the same denominators. • Examples : 1/9, 2/9, 4/9. Lecture Slides By Adil Aslam 72
  73. 73. Like and Unlike Fractions • Unlike fractions are those which have different denominators. • Examples : 2/5, 2/3, 2/9. Lecture Slides By Adil Aslam 73
  74. 74. Lecture Slides By Adil Aslam 74

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