Question

Find the integral linear combination of GCD (29341,1739). GCD = greatest common divisor. (An integer linear combination of a and b is any number of the form ax + by where x and y are also integers. The smallest positive integer linear combination of a and b is GCD(a,b).)

Answer 1

An integer linear combination of two numbers is any number of the form ax + by where x and y are integers. The smallest positive integer linear combination of two numbers is the greatest common divisor (GCD) of the numbers. In this case, we find the GCD of 29341 and 1739 to be 1, so the integral linear combination is 1.

Step-by-step explanation:

An integer linear combination of two numbers is any number of the form ax + by where x and y are integers. The smallest positive integer linear combination of two numbers is the greatest common divisor (GCD) of the numbers. In this case, we need to find the integral linear combination of 29341 and 1739.

We can use the Euclidean algorithm to find the GCD of 29341 and 1739. Let's divide 29341 by 1739:

29341 = 16 * 1739 + 557

Now let's divide 1739 by 557:

1739 = 3 * 557 + 68

Continuing this process:

557 = 8 * 68 + 53

68 = 1 * 53 + 15

53 = 3 * 15 + 8

15 = 1 * 8 + 7

8 = 1 * 7 + 1

7 = 7 * 1 + 0

Since the remainder is 0, the GCD of 29341 and 1739 is 1.

Therefore, the integral linear combination of 29341 and 1739 is 1. We can write this as 29341(557) - 1739(29341) = 1.