6. Show work **Title: Calculating the Greatest Common Divisor Using the Euclidean Algorithm****Problem Statement:**6. Use the Euclidean algorithm to find \(\text{gcd}(825, 315)\).**Solution:**The Euclidean algorithm is an efficient method for computing the greatest common divisor (GCD) of two numbers. It is based on the principle that the GCD of two numbers also divides their difference. To find the GCD using the Euclidean algorithm, follow these steps:1. **Divide 825 by 315** to find the remainder. - \(825 \div 315 = 2\) remainder \(195\) - Expressed in terms of GCD: \(\text{gcd}(825, 315) = \text{gcd}(315, 195)\)2. **Divide 315 by 195**. - \(315 \div 195 = 1\) remainder \(120\) - Update the expression: \(\text{gcd}(315, 195) = \text{gcd}(195, 120)\)3. **Divide 195 by 120**. - \(195 \div 120 = 1\) remainder \(75\) - Update the expression: \(\text{gcd}(195, 120) = \text{gcd}(120, 75)\)4. **Divide 120 by 75**. - \(120 \div 75 = 1\) remainder \(45\) - Update the expression: \(\text{gcd}(120, 75) = \text{gcd}(75, 45)\)5. **Divide 75 by 45**. - \(75 \div 45 = 1\) remainder \(30\) - Update the expression: \(\text{gcd}(75, 45) = \text{gcd}(45, 30)\)6. **Divide 45 by 30**. - \(45 \div 30 = 1\) remainder \(15\) - Update the expression: \(\text{gcd}(45, 30) = \text{gcd}(30, 15)\)7. **Divide 30 by 15**. - \(30 \div 15 = 2\) remainder \(0\)When

6. Use the Euclidean algorithm to find gcd825, 315.The Euclidean algorithm is as shown: Consider 825=2×315+195, Consider 315=1×195+120, Consider 195=1×120+75, Consider 120=1×75+45, Consider 75=1×45+30, Consider 45=1×30+15, Consider 30=2×15+0. Each step is a=bq+r, where a is divided by b where q is the quotient and r is the remainder. The algorithm stops when the remainder is 0. Thus the greatest common divisor is 15. Hence, gcd825, 315=15.

Answered: 6. Use the Euclidean algorithm to find…

Math

Advanced Math

6. Use the Euclidean algorithm to find gcd (825, 315).

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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