I m trying to find a really quick method to calculate stuff like

$x! \pmod y~~ \text{where }~\{x,y:\text{y is a prime}\}\in \mathbb{N}$ and $400\geq \{x,y\}\geq 50$ and $y>x+30$
$\text{Note : }$ Four options are given and the time limit to find is on an average $5$ minutes per question in exam. You can use any method (solution based, not procees based). Pen and Paper method. No calculators allowed.

I know there is a $\text{Wilson's theorem}$ but cannot help every time as here as the difference between $x$ and $y$ can be little more.

example :

$73! \pmod {113}$

Here $112! \pmod{113}\equiv-1$

How to proceed


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