1. find all natural numbers $n>1$ such that $(n-1)$ ! is divisible by $n$.
## 1st solution:

We call $n$ good if $n$ is a divisor of $(n-1)$ ! Let $p$ be the smallest prime divisor of $n$. Then the natural number $\frac{n}{p}$ is either equal to 1 or equal to $p$ or greater than $p$. We now distinguish these three $\stackrel{p}{F}$ cases.

- If $n=p$ is prime, then each of the numbers $1,2, \ldots,(n-1)$ is not related to $n$, so their product ( $n-1$ ) is too! Thus $n$ is bad.
- If $n=p^{2}$ with $p \geq 3$, then $pp$, then $p1$ are good, except the prime numbers and $n=4$.