Question

If a number n has exactly 7 divisors, how many divisors does n squared have?

Answer 1

A number $n=p_1^{a_1}\cdot p_2^{a_2}\cdot\ldots\cdot p_k^{a_k}$, where $p_k$ are distinct prime numbers, has $(a_1+1)(a_2+1)\cdot\ldots\cdot(a_k+1)$ divisors.

We know that $n$ has 7 divisors. 7 is a prime number, so among the numbers $a_1+1,a_2+1,\ldots,a_k+1$ there is only one that is equal to 7.

So,
$a_k+1=7\\ a_k=6$

That means $n=p^6$ where $p$ is a prime number.
$n^2=(p^6)^2=p^{12}$, therefore the $n^2$ has $12+1=13$ divisors.