Question

Prove that 7 cannot be expressed as an integral linear combination of 29341 and 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

To prove that 7 cannot be expressed as an integral linear combination of 29341 and 1739, we need to show that the greatest common divisor (GCD) of 29341 and 1739 is not equal to 1. The GCD of 29341 and 1739 is 1, therefore 7 cannot be expressed as an integral linear combination of 29341 and 1739.

Step-by-step explanation:

To prove that 7 cannot be expressed as an integral linear combination of 29341 and 1739, we need to show that the greatest common divisor (GCD) of 29341 and 1739 is not equal to 1. If the GCD is not equal to 1, then 7 cannot be represented as an integral linear combination of the two numbers. To find the GCD of 29341 and 1739, we can use the Euclidean algorithm:

1. Divide 29341 by 1739: 29341 ÷ 1739 = 16 with a remainder of 757.
2. Divide 1739 by 757: 1739 ÷ 757 = 2 with a remainder of 225.
3. Divide 757 by 225: 757 ÷ 225 = 3 with a remainder of 82.
4. Divide 225 by 82: 225 ÷ 82 = 2 with a remainder of 61.
5. Divide 82 by 61: 82 ÷ 61 = 1 with a remainder of 21.
6. Divide 61 by 21: 61 ÷ 21 = 2 with a remainder of 19.
7. Divide 21 by 19: 21 ÷ 19 = 1 with a remainder of 2.
8. Divide 19 by 2: 19 ÷ 2 = 9 with a remainder of 1.
9. Divide 2 by 1: 2 ÷ 1 = 2 with a remainder of 0.

The GCD of 29341 and 1739 is 1 since we reached a remainder of 0. Therefore, 7 cannot be expressed as an integral linear combination of 29341 and 1739.