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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.

redneckmathematician
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    Could you please clarify precisely what law your refer to, and where you are having trouble. The distributive law $\ a\gcd(b,c) = \gcd(ab,ac)\ $ has been discussed here many times in the past, e.g. I gave four proofs here. – Bill Dubuque Feb 26 '19 at 21:33

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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.

John Omielan
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