Does Wilson's Theorem Help?
Find the largest possible value of such that, when is divided by 2017, it gives a remainder of 1.
Notation:
is the
factorial
notation. For example,
.
The answer is 2015.
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1 solution
If n ≥ 2 0 1 7
2 0 1 7 ∣ n ! ⟹ 2 0 1 7 ∤ n ! − 1
If n = 2 0 1 6 , by Wilson's theorem
2 0 1 6 ! ≡ − 1 m o d 2 0 1 7 ⟹ 2 0 1 6 ! − 1 ≡ − 2 m o d 2 0 1 7 ⟹ 2 0 1 7 ∤ 2 0 1 6 ! − 1
If n = 2 0 1 5
2 0 1 6 ! ≡ 2 0 1 6 m o d 2 0 1 7 ⟹ g cd ( 2 0 1 6 , 2 0 1 7 ) = 1 2 0 1 5 ! ≡ 1 m o d 2 0 1 7 ⟹ 2 0 1 5 ! − 1 ≡ 0 m o d 2 0 1 7 ⟹ 2 0 1 7 ∣ 2 0 1 5 ! − 1
Thus, 2 0 1 5 is the largest number that satisfies the constraint