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What is the remainder when $45!$ is divided by $47$ ?

Is there any method to approach such questions ?


marked as duplicate by user223391, Namaste discrete-mathematics Dec 29 '16 at 18:53

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  • $\begingroup$ What are your thoughts, Garrick? $\endgroup$ – Namaste Dec 29 '16 at 18:51
  • $\begingroup$ @amWhy I guess Wilson Theorem but I don't know how to apply ? $\endgroup$ – Jon Garrick Dec 29 '16 at 18:52

By Wilson's theorem, $46! \equiv -1 \pmod{47}$. Thus, we have $$ -1 \equiv 46! \equiv 46 \cdot 45! \equiv (-1) \cdot 45! \pmod{47} $$ deduce that $45! \equiv 1 \pmod{47}$.

  • $\begingroup$ @amWhy I understand your frustration. If I see a question that looks familiar, I'll look for an older question; however, this was a question I had never seen that had a quick answer, so I just went for it. I'll be more careful in the future. $\endgroup$ – Omnomnomnom Dec 29 '16 at 19:17
  • $\begingroup$ I do understand. Thanks for the reply! $\endgroup$ – Namaste Dec 29 '16 at 19:24

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