Is there a pattern?
I have no idea how to create, just the operation of some divisibility criteria ...
How difficult is it to create an even and odd number criterion if you have one? Is there a greater difficulty if you are cousin or not?
For example: how to create a divisibility criterion for $251$ (prime number) and $342$
Following the method in this answer
$\!\!\bmod 251\!:\ {-}25(10)\equiv 1\ $ so $\,10^{-1}\equiv -25$
hence $\, 10b+a\equiv 0\! \overset{\times 10^{\large -1\!\!\!}}\iff b-25a\equiv 0,\ $ i.e. $\,\bbox[5px,border:1px solid #c00]{251\mid 10b+a\iff 251\mid b-25a}$
e.g. $\ 251\mid 753\iff 251\mid 75-25(3) = 0$
Similarly $\ 171\mid 10b+a\iff 171\mid b-17a\,$ which - with an even test - yields your $342$ test.