Question 145
Let $n$ be a positive integer. Prove that there exists a positive integer $m$ such that $n! = \varphi(m)$, where $\varphi$ is Euler's totient function. (Recall that if $m = p_{1}^{\alpha_{1}} \cdots p_{k}^{\alpha_{k}}$ with $p_{1}, \ldots, p_{k}$ pairwise distinct primes, then $\varphi(m) = m \left(1 - \frac{1}{p_{1}}\right) \cdots \left(1 - \frac{1}{p_{k}}\right)$.)