A number $n$ has $3$ divisors. How many divisors does $n^2$ have?
I need to find a relationship between the divisors of $n$ and the divisors of $n^2$. One way to do that is to use prime factorization. If $n$ has only $3$ divisors, then it must be of the form $p^2$ for some prime $p$, since any other factorization would give more divisors. For example, $4$ has $3$ divisors: $1, 2, 4$, but $6$ has $4$ divisors: $1, 2, 3, 6$. So, $n = p^2$ and $n^2 = p^4$. Now, I can use the formula for the number of divisors of a power of a prime. If $m = q^k$ for some prime $q$, then the number of divisors of $m$ is $k + 1$, since the divisors are $q^0, q^1, \dots, q^k$. So, the number of divisors of $n^2 = p^4$ is $4 + 1 = 5$. 