Number Theory Factor Challenge!!!

Question

For a positive integer $n$ let $P(n)$ denote the product of its positive divisors. E.g. $P(10)=1\times2\times5\times10=100$. Find all solutions to the equation $P(n)=15n^2$.

Answer 1

Divisors come in pairs. More precisely the divisors of a number $n$ that is not a perfect square come in pairs whose product is $n$, and a perfect square has one additional divisor, $\sqrt n$. In both cases, we get a factor $\sqrt n$ per divisor, so $P(n)=n^{\frac d2}$ if $n$ has $d$ divisors. Thus, in your question, $n^{\frac d2}=15n^2$, so $n^{\frac d2-2}=15$. Since $15$ is not a perfect power, you only have to check the values of $d$ with $0\lt\frac d2-2\le1$. Since neither $d=5$, $n=225$, nor $d=6$, $n=15$ works, the equation has no solutions.