Can you help, Mr Wilson?

Number Theory Level 5

Find the remainder when ( 1994 997 ) \displaystyle \dbinom{1994}{997} is divided by 99 7 3 997^{3} .

You may use the fact that 997 is prime .

Notation : ( M N ) \binom MN denotes the binomial coefficient , ( M N ) = M ! N ! ( M − N ) ! \binom MN = \frac{M!}{N!(M-N)!} .


The answer is 2.

Harsh Shrivastava by Harsh Shrivastava , 20, India

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1 solution

Otto Bretscher
Apr 16, 2016

Wolstenholme's Theorem states that ( 2 p − 1 p − 1 ) ≡ 1 ( m o d p 3 ) {2p-1 \choose p-1}\equiv 1 \pmod{p^3} for primes p > 3 p>3 so ( 2 p p ) ≡ 2 {2p \choose p}\equiv \boxed{2}

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