Problem 10RQ

a) How can you find a linear combination (with integer coefficients) of two integers that equals their greatest common divisor?

b) Express gcd(84. 119) as a linear combination of 84 and 119.

Solution:

Step1

a) How can you find a linear combination (with integer coefficients) of two integers that equals their greatest common divisor?

A linear combination (with integer coefficients) of two integers that equals their greatest common divisor is found by Euclidean algorithm.

The Euclidean algorithm, is a productive technique for figuring the best regular divisor(GCD) of two numbers, the biggest number that partitions them two without leaving a remainder.It depends on the rule that the best normal divisor of two numbers does not change if the bigger number is supplanted by its distinction with the more modest number.

Step2

b) Express gcd(84. 119) as a linear combination of 84 and 119.

By using Euclidean algorithm

Therefore, gcd(84,119)=7

By going the inverse directions of the above divisions, we get

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So, Linear combination is