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The question is as follows:

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So I started by attempting to calculate 12^100 mod 47 by modular exponentiation, which if I'm not mistaken, turned out to be:

12^64 * 12^16 * 12^16 * 12^4 (mod 47) = 37 * 28 * 28 * 9 (mod 47) = 34.

So the original expression becomes 34n + 45! = 13 mod 47.

Then, I attempted to work out 45! mod 47 using Wilson's Theorem:

46! = -1 mod47.

I am unsure where to go from here. Would it be correct to divide by 46 to get 45! = -1/46 mod 47?

i.diazr
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  • The first two dupes show how to compute the factorial and power, and the third shows how to solve the resulting linear congruence. In the first with $h=1$ we get $(p-2)! \equiv 1\pmod p,,$ which is also easy to prove directly using $p-1\equiv -1$ in Wilson's formula. – Bill Dubuque Nov 22 '23 at 21:44

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