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Let $a$ and $b$ be positive integers where $a$ is even and such that $\gcd(2a, 2b) = 70$. Find $\gcd(a, 2b)$.

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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?

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  • $\begingroup$ I wanna say that we have gcd(70k,35n) = 70 implies gcd(2k,n) = 2... but I really don't see how to proceed $\endgroup$
    – user600210
    May 15 '19 at 18:02
  • $\begingroup$ You don't really need to know anything about k or n except that they apparently have no factors in common. Think about what the question is asking you to find. $\endgroup$
    – ConMan
    May 15 '19 at 23:34

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