Find all the numbers $n$ such that $\varphi(n)=5$
Attempt:
$$n=\prod\limits_{i=1}^{k} p_i^{\beta_i}$$
$$\varphi(n)=\prod\limits_{i=1}^{k}p^{\beta_i-1}(p-1)$$
We need:
$$\prod\limits_{i=1}^{k}p^{\beta_i-1}(p-1)=5$$
Since$$\varphi(n)\leqslant n-1$$
there isn't such $n$ is this correct?