elementary number theory - Prove that $gcd(2^{a}+1,2^{gcd(a,b)}-1)=1$

Let $a$ and $b$ be two odd positive integers. Prove that $\gcd(2^{a}+1,2^{\gcd(a,b)}-1)=1$.

I tried rewriting it to get $\gcd(2^{2k+1}+1,2^{\gcd(2k+1,2n+1)}-1)$, but I didn't see how this helps.