Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$.
I was thinking among the lines of:
Since $a$ and $b$ are coprime numbers, $\gcd(a,b)=1$. Then there exist integers $x$ and $y$ such that, $ax+by=1$.
Then, $a=(1-by)/x$, $b=(1-ax)/y$
So if I write $2^a+1$ as:
$2^{(1-by)/x}+1$
Then can I say that the above expression is equivalent to 0 (mod 3)?