DE-BUNDESWETTBEWERB 2004 Problem 5

Let $k$ be a positive integer. A natural number $m$ is called $k$ -typical if each divisor of $m$ leaves the remainder $1$ when being divided by $k$ .

Prove:

a) If the number of all divisors of a positive integer $n$ (including the divisors $1$ and $n$ ) is $k$ -typical, then $n$ is the $k$ -th power of an integer.

b) If $k > 2$ , then the converse of the assertion a) is not true.