Dividing by 2017

Number Theory Level 3

What is the remainder when 2016 ! 2016! is divided by 2017?

Notation: ! ! denotes the factorial notation . For example 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1 \times 2 \times 3 \times \cdots \times 8 .

2016 1 0 1008 2 2015

William Steinberg by William Steinberg , 20, Australia

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1 solution

Chew-Seong Cheong
May 14, 2017

By Wilson's theorem , if p p is a prime, then ( p − 1 ) ! ≡ − 1 ( m o d p ) (p-1)! \equiv -1 \pmod p holds. Since 2017 is a prime, 2016 ! ≡ − 1 ≡ 2016 ( m o d 2017 ) 2016! \equiv -1 \equiv \boxed{2016} \pmod {2017} .

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