Let d=gcd(a+b,a*b)

Then d divides (a+b)x+(a*b)y, where x and y can be any integers.

If you let x=a and y=-1, you get : a²+ab-ab=a². Thus d divides a²

If you let x=b and y=-1, you get : ab+b²-ab=b². Thus d divides b²

---> d=1.

d divides (a+b) and (a-b) and thus divides (a+b)+(a-b)=2a and (a+b)-(a-b)=2b.2) let gcd(a,b) = 1. prove that d= gcd(a+b, a-b) = 1 or 2

(hint: prove d<2)

Thanks!

So d divides 2 or d divides a. If it doesn't divide 2, it divides a and b. Then...

If it divides 2, then...