I was just playing with the factorial and the modulo function. I just observed this interesting property. I was using a calculator
$$13!\equiv 13\times 12\pmod{169}\\ 17!\equiv 17\times 16\pmod{289}$$
It is easily verifiable that this works for $2,3,5,7,11$ also.
I conjecture that for any prime $p$, $$p!\equiv (p)\times (p-1)\pmod{p^2}$$
How does one go about proving it? and by the way is this well known or anything?
By Wilson's Theorem we have $(p-1)!\equiv -1\equiv p-1\pmod{p}$. Your conjectured result is obtained by multiplying through by $p$.