Question
Fri March 01, 2013 By:

Prove that g.c.d. (a-b, a+b) = 1 or 2, if g.c.d. (a,b) = 1

Expert Reply
Sat March 02, 2013
Let d = gcd(a+b, a-b) > 0. We will show that d = 1 or 2.

Since d is a divisor of a+b and a-b, it is a divisor of their sum as well as difference, 
sum : (a+b) + (a-b) = 2a
difference: (a+b) - (a-b) = 2b.

Thus, since d is a divisor of both 2a and 2b, and yet the gcd of a and b is 1, then we're left with d = 1 or 2.
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