When is the number of divisors odd?

Pairing. For each divisor $d$ of $n$, $n/d$ is also a divisor. The map $d\mapsto n/d$ is an involution. Fixed points satisfy $d=n/d$, i.e. $d^2=n$.

Conclusion. If $n$ is not a square, all divisors pair up, so the number of divisors is even. If $n=k^2$, all divisors pair up except $d=k$, so the number of divisors is odd.