This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it.
So anyway, here the problem goes:
The $\text{Factof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\text{Factof}(18) = \frac{18}{6} = 3$, and $27$ has $4$ factors so $\text{Factof}(27) = \frac{27}{4} = 6.75$.
Show that the square of any prime number is the $\text{Factof}$ of some integer.
Edit: I have reworded the question so that it is more specific and can be answered coherently in the context of the rules.
Best Answer
Given the prime $p\neq 3$, the integer number you are looking for is $n(p) = 3^2p^2$. It has $3^2 = 9$ factors ($1$, $3$, $p$, $3p$, $3^2p$, $3p^2$, $3^2$, $p^2$, $3^2p^2$) and $$ \mbox{Factof}(n(p)) = \frac{3^2p^2}{3^2} = p^2 $$ If $p=3$ then the number $n(p)=2^2 3^3 = 108$. In this case
$$ \mbox{Factof}(108) = \frac{2^23^3}{2^2 3} = 3^2 $$