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?

The prime factorization of $XY$ is the same as that of $X$ times that of $Y$. It is clear that it won't share any factors with $Z$. — AvZ, Jan 29 '15 at 17:40
@AvZ that's true but sometimes the result is proved as a preparation to show uniqueness of prime factorization, so this might not be available. — quid, Jan 29 '15 at 17:48

score 2 · Answer 1 · answered Jan 29 '15 at 17:41

2

aX + bZ = 1 and cY + dZ = 1

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

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

Therefore etc.

answered Jan 29 '15 at 17:41

So if you can write $1=ax+by$ then $a$ and $b$ are coprime? I.e it is an if and only if situation? – Jared Jan 29 '15 at 17:52
Yes it is; this is why it's very useful. – Jan 29 '15 at 17:53
Ah I see okay thanks! – Jared Jan 29 '15 at 17:54

score 0 · Answer 2 · answered Jan 29 '15 at 17:39

0

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

answered Jan 29 '15 at 17:39

Casteels

2 Answers2