Contrapositive Proof to show that for $x,y ∈ ℤ$, if $5∤xy$, then $5∤x$ and $5∤y$

Question

I need to use a contrapositive proof to show that for $x,y ∈ ℤ$, if $5∤xy$, then $5∤x$ and $5∤y$.

So far I've got that the contrapositive statement would be "if $5|xy$, then $5|x$ and $5|y$". Obviously this is not true as $2*5=10$, 10 and 5 being divisible by five but 2 is not.

If xy is divisible by 5, there must exist some number a ∈ ℤ such that $xy = 5a$.

If x is divisible by 5, there must exist some number b ∈ ℤ such that $x = 5b$.

If y is divisible by 5, there must exist some number c ∈ ℤ such that $y = 5c$.

$5a = 5b * 5c$

$5a = 25bc$

$a = 5bc$

Any help would be greatly appreciated, I've spent days spinning my wheels on this.

Answer 1

The correct contrapositive for your statement is if 5|x or 5|y then 5|xy.