If $x$ is a positive integer, what is the value of $x$ for the equation $(x!-(x-3)!) \div 23 = 1$?
We have $x!-(x-3)! = 23$. Since $4!=24$, the number $23$ strongly suggests us to try $x=4$, and indeed, $4!-(4-3)! = 4!-1! = 24-1=23$, so $x=\boxed{4}$ is the answer.