Modulus 1009 #1

Number Theory Level 3

Find the remainder when ( 1011 ) ! 1009 \frac{(1011)!}{1009} is divided by 1009

NOTE: \textbf{NOTE:} 1009 is a prime number


The answer is 1007.

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Ivan Molina N by Ivan Molina N , 28, Colombia

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1 solution

We have to find the remainder when ( 1011 ) ! 1009 \dfrac{{(1011)!}}{{1009}} or 1008 ! × 1010 × 1011 1008! \times 1010 \times 1011 is divided by 1009.

By Wilson's theorem ( P − 1 ) ! + 1 ≡ 0 ( m o d P ) (P - 1)! + 1 \equiv 0(modP) .

So 1008 ! + 1 = 0 m o d 1009 1008! + 1 = 0 mod 1009

or 1008 ! = − 1 m o d 1009 1008! = - 1 mod1009

Therefore, the required remainder = − 1 × 1 × 2 = − 2 - 1 \times 1 \times 2 = - 2 or 1007

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