Euler's totient theorem states that if $a$ and $n$ are coprime positive integers, then: $$a^{\varphi(n)} \equiv 1\;(\mathrm{mod}\;n)$$ So if $k$ is a given positive integer, which satisfies: $$\begin{cases} a^k \equiv 1\;(\mathrm{mod}\;n)\\ k\leqslant \varphi(n) \end{cases}$$
Does it necessarily hold that $k\,|\,\varphi(n)$?