Claims: If \(n\) is coprime with \(a\) and \(b\), then \(n\) is coprime with \(ab\).
Firstly let us define what we mean by coprime. If \(n\) and \(a\) are coprime, then
$$ gcd(n, a) = 1 $$
Using Euclidean algorithm, we know that if (and only if) the above is true then there must exist integers \(x\) and \(y\) such that
$$ ax + ny = 1 $$
So then to prove the result above, let \(v\), \(w\), \(x\), \(y\) be some integer such that
$$\begin{aligned}
av + nw & = 1 \\
bx + ny & = 1
\end{aligned}$$
Multiplying these two equations together yields
$$ vx(ab) + n(avy + bxw + nwy) = 1 $$
Thus there exists an integer linear combination of \(ab\) and \(n\), so \(gcd(ab, n) = 1\) and \(ab\) must be coprime with \(n\).