Integers Representable as Product of both 3 and 4 Consecutive Integers

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: