Integers Representable as Product of both 3 and 4 Consecutive Integers

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


 * $24, 120, 175 \, 560$

Proof
We have: