Problem

Source: IMO ShortList 1998, number theory problem 6

Tags: number theory, Divisors, Number theoretic functions, prime factorization, IMO, IMO 1998, IMO Shortlist



For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.