Smallest Numbers with 240 Divisors

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Theorem

The smallest integers with $240$ divisors are:

$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$


Proof

In the below, $\map {\sigma_0} n$ denotes the divisor count function of $n$.


Then:

\(\ds \map {\sigma_0} {720 \, 720}\) \(=\) \(\ds 240\) $\sigma_0$ of $720 \, 720$
\(\ds \map {\sigma_0} {831 \, 600}\) \(=\) \(\ds 240\) $\sigma_0$ of $831 \, 600$
\(\ds \map {\sigma_0} {942 \, 480}\) \(=\) \(\ds 240\) $\sigma_0$ of $942 \, 480$
\(\ds \map {\sigma_0} {982 \, 800}\) \(=\) \(\ds 240\) $\sigma_0$ of $982 \, 800$
\(\ds \map {\sigma_0} {997 \, 920}\) \(=\) \(\ds 240\) $\sigma_0$ of $997 \, 920$



Sources