# Are all highly composite numbers even?

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.

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.