Let $a$ and $b$ be positive integers where $a$ is even and such that $\gcd(2a, 2b) = 70$. Find $\gcd(a, 2b)$.
If $\gcd(2a, 2b) = 70$ then we know that $2a = 70m$ and $2b = 70n$ for some integers $m, n$ (where $\gcd(m, n) = 1$), which means that $a = 35m$. But, since $a$ is even, $a = 70k$ for some integer $k$ (where $m = 2k$).
Can you go from there?