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.
2026-03-26 03:18:44.1774495124
Are all highly composite numbers even?
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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.