Question

Find the remainder when 45 factorial is divided with 47?

Answer 1

By Wilson's Theorem, if $\sf{p}$ is a prime number, we have,

$\longrightarrow\sf{(p-1)!\equiv-1\pmod{p}}$

$\longrightarrow\sf{(p-2)!\,(p-1)\equiv-1\pmod{p}}$

Since $\sf{p-1\equiv-1\pmod{p},}$

$\longrightarrow\sf{-(p-2)!\equiv-1\pmod{p}}$

$\longrightarrow\sf{(p-2)!\equiv1\pmod{p}}$

Let $\sf{p=47}$ since it's a prime number. Therefore,

$\longrightarrow\sf{(47-2)!\equiv1\pmod{47}}$

$\longrightarrow\underline{\underline{\sf{45!}\equiv\bf{1}\pmod{\sf{47}}}}$

Hence 1 is the remainder.