If $X$ and $Y$ are coprime to $Z$, then so is their product $XY$

Question

Given is $X$ is coprime to $Z$ and $Y$ is coprime to $Z$ prove $XY$ is coprime to $Z$.

I know you can use Bezout's lemma to say $1=aX+bZ$ and $1=cY+dZ$ but I don't know how to actually do the proof.

Any ideas?

Answer 1

Hint: If $XY$ and $Z$ are not coprime, then there's a prime number dividing both $XY$ and $Z$...

Answer 2

aX + bZ = 1 and cY + dZ = 1

=> (aX + bZ)(cY + dZ) = 1

=> (ac)XY + (adX + bcY + bdZ)Z = 1

Therefore etc.