Integers Representable as Product of both 3 and 4 Consecutive Integers

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Theorem

There are $3$ integers which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$:

$24, 120, 175 \, 560$


Proof

We have:

\(\ds 24\) \(=\) \(\ds 1 \times 2 \times 3 \times 4\)
\(\ds \) \(=\) \(\ds 2 \times 3 \times 4\)
\(\ds 120\) \(=\) \(\ds 2 \times 3 \times 4 \times 5\)
\(\ds \) \(=\) \(\ds 4 \times 5 \times 6\)
\(\ds 175 \, 560\) \(=\) \(\ds 55 \times 56 \times 57\)
\(\ds \) \(=\) \(\ds 19 \times 20 \times 21 \times 22\)



Sources