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Monday, February 22, 2016

2016/016) Show that 4 * (29!) + 5! \equiv 0 (\,mod\, 31)

because 31 is prime we have as per wilson's theorem
30!  \equiv -1 (\,mod\, 31)\cdots(1)
and also 30 * (-1) = -30  \equiv 1 (\,mod\, 31) =>30^{-1} = \equiv 1 (\,mod\, 31)\cdots(2)
from (1) and (2)
29!  \equiv 1 (\,mod\, 31)
or 4 * 29!  \equiv 4 (\,mod\, 31)
or 4 * 29! + 5!   \equiv 4 + 120 (\,mod\, 31) \equiv  124 (\,mod\, 31) \equiv 0 (\,mod\, 31)

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