A simple question about divisibility: If $n$ divides the product of coprime numbers...

Question

This is quite basic, but I'm no good at this stuff, and I've basically just been sitting her trying to find counterexamples...

If $n$ divides $jk$, where $gcd(j,k)=1$, must $n$ divide $j$ and $k$? Or, perhaps, n must divide $j$ or $k$? Or neither?

Answer 1

Hint $6$ divides $4 \cdot 9$...

P.S. If $n |jk$ with $gcd(j,k)=1$, what is true is that we can write $n=d_1d_2$ with $d_1|j, d_2|k$ in an unique way. This fact is often used in number theory. The existence comes from $n|jk$, it is the uniqueness which is guaranteed by the relatively prime conditions (but uniqueness doesn't necessarily imply relatively primeness of $k,j$,)

Answer 2

Let $j=\prod_i p_i^{\alpha_i}$ and $k=\prod_j q_i^{\alpha_j}$, where $p_i,q_i$ are distinct primes, so that $\gcd(i,j)=1$.

Any $n$ that is a non-trivial combination of powers of $p_i's$ and $q_j's$ divides $ij$ but neither $i$ or $j$.