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Use the Euclidean Algorithm to compute the greatest common divisors. 1. gcd(291,252)

Use the Euclidean Algorithm to compute the greatest common divisors. 1. gcd(291,252)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Euclidean Algorithm for Computing Greatest Common Divisors The Euclidean Algorithm is a method for finding the greatest common divisor (GCD) of two numbers. The GCD of two integers is the largest positive integer that divides both of the integers without leaving a remainder. Below are a few examples of how to use the Euclidean Algorithm to compute the GCD. #### Examples: 1. **gcd(291, 252)** 2. **gcd(16261, 85652)** 3. **gcd(139024789, 93278890)** 4. **gcd(16534528044, 8332745927)** ### Steps to Compute GCD Using the Euclidean Algorithm: 1. **Divide** the larger number by the smaller number. 2. **Replace** the larger number with the smaller number and the smaller number with the remainder of the division. 3. **Repeat** steps 1 and 2 until the remainder is zero. 4. The last non-zero remainder is the GCD of the two numbers. For example, to compute gcd(291, 252): - Divide 291 by 252, which gives a quotient of 1 and a remainder of 39 (since 291 = 252 * 1 + 39). - Replace 291 with 252 and 252 with 39. Now find gcd(252, 39). - Divide 252 by 39, which gives a quotient of 6 and a remainder of 18 (since 252 = 39 * 6 + 18). - Replace 252 with 39 and 39 with 18. Now find gcd(39, 18). - Divide 39 by 18, which gives a quotient of 2 and a remainder of 3 (since 39 = 18 * 2 + 3). - Replace 39 with 18 and 18 with 3. Now find gcd(18, 3). - Divide 18 by 3, which gives a quotient of 6 and a remainder of 0 (since 18 = 3 * 6 + 0). - The remainder is now 0, so the GCD is the last non-zero remainder, which is 3. This method efficiently helps find the GCD for any pair of positive integers.
Transcribed Image Text:### Euclidean Algorithm for Computing Greatest Common Divisors The Euclidean Algorithm is a method for finding the greatest common divisor (GCD) of two numbers. The GCD of two integers is the largest positive integer that divides both of the integers without leaving a remainder. Below are a few examples of how to use the Euclidean Algorithm to compute the GCD. #### Examples: 1. **gcd(291, 252)** 2. **gcd(16261, 85652)** 3. **gcd(139024789, 93278890)** 4. **gcd(16534528044, 8332745927)** ### Steps to Compute GCD Using the Euclidean Algorithm: 1. **Divide** the larger number by the smaller number. 2. **Replace** the larger number with the smaller number and the smaller number with the remainder of the division. 3. **Repeat** steps 1 and 2 until the remainder is zero. 4. The last non-zero remainder is the GCD of the two numbers. For example, to compute gcd(291, 252): - Divide 291 by 252, which gives a quotient of 1 and a remainder of 39 (since 291 = 252 * 1 + 39). - Replace 291 with 252 and 252 with 39. Now find gcd(252, 39). - Divide 252 by 39, which gives a quotient of 6 and a remainder of 18 (since 252 = 39 * 6 + 18). - Replace 252 with 39 and 39 with 18. Now find gcd(39, 18). - Divide 39 by 18, which gives a quotient of 2 and a remainder of 3 (since 39 = 18 * 2 + 3). - Replace 39 with 18 and 18 with 3. Now find gcd(18, 3). - Divide 18 by 3, which gives a quotient of 6 and a remainder of 0 (since 18 = 3 * 6 + 0). - The remainder is now 0, so the GCD is the last non-zero remainder, which is 3. This method efficiently helps find the GCD for any pair of positive integers.
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