Distributive law of greatest common divisors.

Question

I have seen someone use a distributive law of gcds but I was wondering if anybody could prove that as I am having a little trouble going about this.

Answer 1

By the "distributive law of greatest common divisors", I assume you mean something like proving that $\gcd(na,nb) = n \cdot \gcd(a,b)$ where $n$, $a$ and $b$ are all natural numbers. If so, this is asked & there are $4$ answers to this at the MSE page elementary number theory - GCD Proof with Multiplication: gcd(ax,bx) = x$\cdot$gcd(a,b). A comment says it's the same as How to prove that $z\gcd(a,b)=\gcd(za,zb)$, and I wouldn't be surprised if there are several other places where this has already been dealt with in MSE.

If you mean something else, please clarify what you are looking for. Thanks.