Find the remainder when 45 factorial is divided with 47?
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Find the remainder when 45 factorial is divided with 47?
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By Wilson's Theorem, if [tex]\sf{p}[/tex] is a prime number, we have,
[tex]\longrightarrow\sf{(p-1)!\equiv-1\pmod{p}}[/tex]
[tex]\longrightarrow\sf{(p-2)!\,(p-1)\equiv-1\pmod{p}}[/tex]
Since [tex]\sf{p-1\equiv-1\pmod{p},}[/tex]
[tex]\longrightarrow\sf{-(p-2)!\equiv-1\pmod{p}}[/tex]
[tex]\longrightarrow\sf{(p-2)!\equiv1\pmod{p}}[/tex]
Let [tex]\sf{p=47}[/tex] since it's a prime number. Therefore,
[tex]\longrightarrow\sf{(47-2)!\equiv1\pmod{47}}[/tex]
[tex]\longrightarrow\underline{\underline{\sf{45!}\equiv\bf{1}\pmod{\sf{47}}}}[/tex]
Hence 1 is the remainder.