basic coset computation

Question

Let $r$ be a nonzero integer, and let $b=|r|+1$ with $b<n$, where $n$ is a positive integer.

Then, how to show that $n!\not\in r+b\mathbb{Z}$?

It seems to be simple, but I can't find any relation between of them.

Give some comment. Thank you!

Answer 1

Note that $n!$ is the product of all positive integers $k$ with $k\le n$. Since $b$ is such an integer, $n!$ is a multiple of $b$.