Question

30. Two integers a and b that are not both zero are relatively prime whenever. A) (a, b) = 1 C) (a, b)=d>1 B) [a, b] = 1 D) None of these​

Answer 1

Prove that if a and b are relatively prime, then gcd(a+b,a−b)=1 or 2.

I started off by putting gcd(a+b,a−b)=d. This implies that there are two relatively prime integers x1,x2, such that

dx1=a+b

dx2=a−b

Adding the first equation to the second gives us: d(x1+x2)=2a, and subtracting the second from the first gives us d(x1−x2)=2b. This implies that d∣2a,d∣2b⟹gcd(2a,2b)=d⟹d=2⋅gcd(a,b)⟹d=2

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