2000 IMO Shortlist S02

Determine all positive integers $n\geq 2$ that satisfy the following condition: for all $a$ and $b$ relatively prime to $n$ we have $$a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.$$

确定满足以下条件的所有正整数 $n\geq 2$：对于所有 $a$ 和 $b$ 与 $n$ 互质，我们有 $$a \equiv b \pmod n\qquad\text{当且仅当}\qquad ab\equiv 1 \pmod n.$$