Prove $2^a+1\mid 2^{ab}+1$ if $2\nmid b$, $a,b\in\mathbb{Z}^{+}$

Question

Prove that $2^a+1\mid 2^{ab}+1$ if $2\nmid b$, $a,b\in\mathbb{Z}^{+}$.

I tried Euclidean Algorithm: $\gcd(2^a+1,2^{ab}+1)=\gcd(2^a+1,2^{ab}-2^a)=\gcd(2^a+1,2^a(2^b-1))=\gcd(2^a+1,2^b-1)$ but I'm not sure what to do from there.

Also I know of the lemma $\gcd(a^n−1,a^m−1)=a^{\gcd(n,m)}−1$, I'm not sure if it helps at all.

Answer 1

Hint: $$ y^b+1 = (y+1)(y^{b-1} - y^{b-2} + \cdots +y^2-y + 1) $$