Given that $\frac{p}{q} = e = 1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{k!} + \frac{e^{z}}{(k+1)!}$ for some $z$ in $[0,1]$ (using Taylor's theorem), and that $s_k = 1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{k!}$ prove $k!(e-s_k)$ is irrational.
I have no idea how to start this - all hints appreciated.
Hint: if $m,n\in \mathbb Z$ and $a$ is irrational then $m(a-n)$ is irrational because $m(a-n)=r\in\mathbb Q$ implies $a=r/m+mn\in\mathbb Q$.