The document discusses the Euclidean Division Algorithm, which defines a method for finding the gcd of two integers. It highlights the algorithm's efficiency, simplicity, and broad applicability across various fields, including mathematics, computer science, and engineering. The conclusion emphasizes its importance in algorithm design and computational complexity.

1. NAME- RANJAN DAS
STUDENT CODE- BWU/BTA/22/508
DEPT- B.TECH CSE (AIML)
SEC – I
SEM- 4th
COURSE NAME- DISCRETE MATHEMATICS
COURSE CODE- PCC-CSM405
TOPIC – DIVISON ALGORITHM

2. • DESCRIPTION
• EXAMPLE
• ADVANTAGES
• APPLICATIONS
• CONCLUSION
INDEX

3. EUCLID'S DIVISON ALGORITHM
Division algorithm states that given any two integers a & b with b > 0;
There exist unique integers q & r such that a = bq +r, 0<= r<b, q Z.
∈
If r=0 then "b" is the HCF/GCD of "a & b .
Ex- a = 50, b = 8
50 = 8 * 6 + 2
q = 6 , r = 2
a=bq+r
8)50( 6
48
2
If r ≠ 0 , then apply Euclid'sdivision lemma to b and r.
b = (r*m) + n
• For some integers m and n, 0 <= n < r
• Continue this process till the remainder is zero.
• The divisor which gives you the remainder as 0 is your HCF/GCD

6. ADVANTAGES
• EFFICIENCY: THE ALGORITHM IS HIGHLY EFFICIENT IN FINDING THE GREATEST COMMON DIVISOR (GCD)
OF TWO NUMBERS. ITS TIME COMPLEXITY IS PROPORTIONAL TO THE NUMBER OF DIGITS IN THE SMALLER
NUMBER, MAKING IT SUITABLE FOR LARGE INTEGERS.
• SIMPLICITY: THE ALGORITHM IS STRAIGHTFORWARD AND EASY TO UNDERSTAND. IT CONSISTS OF A FEW
BASIC STEPS, MAKING IT ACCESSIBLE EVEN TO THOSE WITH LIMITED MATHEMATICAL BACKGROUND.
• APPLICABILITY: IT HAS BROAD APPLICABILITY ACROSS DIFFERENT FIELDS, INCLUDING MATHEMATICS,
COMPUTER SCIENCE, AND CRYPTOGRAPHY. MANY ALGORITHMS AND MATHEMATICAL PROCEDURES RELY
ON THE EUCLIDEAN DIVISION ALGORITHM AS A FUNDAMENTAL STEP.
• DETERMINISTIC: THE ALGORITHM FOLLOWS A DETERMINISTIC PROCESS, MEANING IT WILL ALWAYS
PRODUCE THE SAME RESULT FOR THE SAME INPUT. THIS PROPERTY IS CRUCIAL FOR ITS RELIABILITY AND
PREDICTABILITY IN VARIOUS APPLICATIONS

7. APPLICATION
• COMPUTER SCIENCE (E.G., FINDING MODULAR INVERSES, CRYPTOGRAPHIC ALGORITHMS)
• MATHEMATICS (E.G., SIMPLIFYING FRACTIONS, FINDING FACTORS)
• ALGORITHM ANALYSIS (EG: SOLVING TIME COMPLEXITY FOR AN ALGORITHM)
• ARCHITECTURE AND CONSTRUCTION.
• COMPUTER GRAPHICS AND ANIMATION.
• MECHANICAL ENGINEERING.
• ROBOTICS AND AUTOMATION.

8. CONCLUSION
• EUCLIDEAN DIVISION ALGORITHM PROVIDES A SYSTEMATIC WAY TO
FIND THE GCD OF TWO NUMBERS.
• IT IS FUNDAMENTAL IN VARIOUS FIELDS, INCLUDING MATHEMATICS AND
COMPUTER SCIENCE.
• ITS DETERMINISTIC NATURE AND ABILITY TO HANDLE LARGE NUMBERS
MAKE IT INDISPENSABLE IN ALGORITHM DESIGN AND ANALYSIS,
SERVING AS A FUNDAMENTAL EXAMPLE IN UNDERSTANDING
COMPUTATIONAL COMPLEXITY AND ALGORITHMIC EFFICIENCY.

9. THANK YOU