[ Divisibility of numbers. General properties ]  [ Number and sum of divisors of a number ]

Prime numbers have only two distinct divisors - one and the number itself. What numbers have only three distinct divisors?
Notice that for any divisor $m$, there corresponds a "paired divisor" $-{ }^{M / m}$.

## Solution

Any number $M$ always has divisors 1 and $M$. If $M$ has a divisor $m$, then it also has a divisor ${ }^{M} / m$. Therefore, for $M$ to have exactly three distinct divisors, the following conditions must be met: $m={ }^{M} / m$ (i.e., $M=m^{2}$) and $m$ is a prime number. Hence, it follows that only the squares of prime numbers have exactly three distinct divisors.

## Answer

The squares of prime numbers.