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.