HMMT 2019 U32

For positive integers $a$ and $b$ such that $a$ is coprime to $b$, define $\operatorname{ord}_{b}(a)$ as the least positive integer $k$ such that $b \mid a^{k}-1$, and define $\varphi(a)$ to be the number of positive integers less than or equal to $a$ which are coprime to $a$. Find the least positive integer $n$ such that

$$\operatorname{ord}_{n}(m)<\frac{\varphi(n)}{10}$$

for all positive integers $m$ coprime to $n$.