Question 126
Let $n\gt 1$ be a positive integer; we denote by $d(n)$ the number of positive divisors of $n$. To compute $d(n)$, it suffices to write the factorization of $n$ into powers of distinct primes and multiply the successors of the exponents. For example, for $2016$ we have the factorization $2016=2^{5} \cdot 3^{2} \cdot 7^{1}$ and $d(2016)=(5+1)(2+1)(1+1)=36$.
a. Prove that if $n$ is a perfect square, then $d(n)$ is odd.
b. Determine all $n$ less than $400$ such that both $n$ and $d(n)$ are perfect squares.
a. Prove that if $n$ is a perfect square, then $d(n)$ is odd.
b. Determine all $n$ less than $400$ such that both $n$ and $d(n)$ are perfect squares.