TITLE: Are all highly composite numbers even?
QUESTION [13 upvotes]: A highly composite number is a positive integer with more divisors than any smaller positive integer. Are all highly composite numbers even (excluding 1 of course)? I can't find anything about this question online, so I can only assume that they obviously are. But I cannot see why.

REPLY [25 votes]: Yes. Given an odd number $n$, choose any prime factor $p$, and let $k\geq 1$ be the number such that $p^k\mid n$ but $p^{k+1}\not\mid n$. Then $n\times\frac{2^k}{p^k}$ has the same number of factors, and is smaller.