Question 117
Initially, a positive integer $N$ is written on the board. At each moment, Misha may choose a number $a \gt 1$ on the board, remove it, and write down all its positive divisors except $a$ instead. After some time it happened that the board contains $N^2$ numbers. Determine all values of $N$ for which this can happen. (M. Fadin, K. Kovalenko)