JMO 2000 Problem 4

Given a natural number $n\ge 3$, prove that there exists a set $A_n$ with the following two properties:

1. $A_n$ consists of $n$ distinct natural numbers
2. For any $a\in A$, the remainder of the product of all elements of $A_n\backslash \{a\}$ divided by $a$ is $1$.