A number theory problem by Sayantan Saha
Number Theory
Level
3
What will be the remainder when is divided by 31?
The answer is 1.
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1 solution
(26)(27)(28)(29)(30) ≡ (-5)(-4)(-3)(-2)(-1) (mod 31) → (26)(27)(28)(29)(30) ≡ -(5!) (mod 31) → 30! ≡ -(25!5!) (mod 31). By Wilson's theorem, we get, 30! ≡ -1 (mod 31) → -(25!5!) ≡ -1 (mod 31) → (25!)(5!) ≡ 1 (mod 31). Hence, the answer is 1.