I hope 2017 can divide N.

Number Theory Level 4

$\large N=\color{#D61F06}{1\cdot1!}+\color{#EC7300}{2\cdot2!}+\color{#CEBB00}{3\cdot3!}+\cdots+\color{#20A900}{2014\cdot2014!}+\color{#3D99F6}{2015\cdot2015!}$

For $N$ as defined above, find the remainder when $N$ is divided by 2017.

Notation: $!$ is the factorial notation ; for example: $8! = 1 \times 2 \times 3 \times \cdots \times 8$ .

The answer is 2015.

1 solution

Kushal Bose
Feb 28, 2017

Relevant wiki: Wilson's Theorem

$N=\sum_{r=0}^{2015} r.r!=\sum_{r=0}^{2015} (r+1-1)r!=\sum_{r=0}^{2015} ((r+1)!-r!)=2016!-1!$

From Wilson's Theorem $2016! \equiv -1 \pmod{2017}$ as $2017$ is a prime number.

So, $N \equiv -1-1 \pmod{2017} \\ N \equiv -2 \pmod{2017} \\ N \equiv 2015 \pmod{2017}$

I hope 2017 can divide N.

The answer is 2015.

1 solution

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