GCD (a + b, a - b) - Math Central

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Question from Tomas, a student:

QUESTION:
Let a and b integer and relatively prime. Prove that:
GCD (a + b , a - b) = 1 or 2
regards

Tomas

Hi Tomas.

Let x = GCD (a + b, a - b).

Then x | (a+b) and x | (a-b), by definition.

Thus, x | [(a+b) + (a-b)] (if it goes into one and it goes into the other, it goes into the sum).

But this reduces to x | 2a. So either x | 2, or x | a. But since a and b are relatively prime, we know that x cannot divide a unless x = 1.

Thus x = 1 or 2.

Hope this helps,
Stephen La Rocque.

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