How to prove that if $a$ and $b$ are relatively prime, then $\gcd(a, c) = \gcd(ab,c)$?
How to make a connection between $(a,b)=1$ and $ab$? I have no idea.
[Math] Proving that if $a$ and $b$ are coprime, then $\gcd(a, c) = \gcd(a,bc)$
divisibilityelementary-number-theory
Best Answer
I presume the problem is $\rm\: \color{#C00}{(a,b)=1}\:\Rightarrow\: (a,bc) = (a,c),\:$ where $\rm\:(x,y) := gcd(x,y).\:$ One quick proof uses basic gcd laws: $\rm\,\ (a,bc) = (a,bc,ac) = (a,(bc,ac)) = (a,\color{#C00}{(b,a)}c) = (a,c)$.
Alternatively: if $\rm\:d\mid a\:$ then $\rm\:d\mid bc\iff d\mid bc,ac\iff d\mid (bc,ac)=\color{#C00}{(b,a)}c = c.\:$ So $\rm\:a,bc\:$ and $\rm\:a,c\:$ have equal sets $\rm\,D\,$ of common divisors $\rm\:d,\:$ so equal greatest common divisor (= max $\rm\,D$).
Remark $\ $ If desired, one can trade gcd laws for Bezout identities, e.g
$$\rm\:(a,b)=1\:\Rightarrow\:\exists\, j,k\in\Bbb Z\!:\ ja\!+\!kb=1\ \ \ so\ \ \ d\mid a,bc\:\Rightarrow\: d\mid (ja\!+\!kb)c = c\qquad $$
which trades the above use of the gcd distributive law for said Bezout identity. However, the proofs using the gcd laws work more generally (general gcd domains need not have linear (Bezout) gcds).