[Math] Prove that if 2 divides n and 7 divides n, then 14 divides n

Question

Okay so I have to prove this. I can write that if 2 divides n and 7 divides n, then there must be integers k and m such that
$2*k=n$
and
$7*m=n$

So $14*k*m=n^2$

But what to do after that?

If I say that then 14 divides $n^2$, I get bit of a circular argument, but if I write that n divides $14*k*m$, then I don't know what to do next.

Any help/suggestions?

Answer 1

Following from what you have written, $$n = 2k=7m \implies k=\frac{7m}{2}.$$ Since $k$ is an integer and $\gcd(2,7)=1$, $m/2$ must be an integer; i.e., $m/2=r \implies m=2r$, where $r$ is an integer. Therefore, $$n=7m=7\times 2r = 14 r.$$ Q.E.D.