[Math] If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a gcd of $3$.

Question

If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a $\gcd$ of $3$.

I'm very new to number theory so please don't assume I am familiar with some of the terminology.

Answer 1

Hint: $ \gcd(a + 2b, 2a + b) = \gcd(a + 2b, (2a + b) - 2(a + 2b)) = \gcd(a + 2b, -3b) $.