show that if a and b are relatively prime then gcdab ab 1 or

show that if a and b are relatively prime, then gcd(a+b, a-b) =1 or 2

Solution

Let gcd(a+b,ab)=d.

This implies that there are two relatively prime integers x,y such that

dx=a+b and

dy=ab

Adding the first equation to the second gives us: d(x+y)=2a, and

subtracting the second from the first gives us d(xy)=2b.

This implies that d|2a,d|2bgcd(2a,2b)=dd=2gcd(a,b)d=2

Since d is a positive integer, d = 1 or 2.