3 Years Later

Number Theory Level 4

What is the remainder left when 2014 ! 2014! is divided by 2017?


The answer is 1008.

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Aaaaa Bbbbb
Jul 8, 2014

It is easy to see that, with N is a prime number: ( N − 1 ) ! m o d ( N ) = N − 1 ⇒ 2016 ! m o d ( 2017 ) = 2016 (N-1)! \mod(N)=N-1 \Rightarrow 2016! \mod(2017)=2016 Because: 2015 × 2016 m o d ( 2017 ) = 2 ⇒ 2014 ! m o d ( 2017 ) = 1008 2015 \times 2016 \mod(2017)=2 \Rightarrow 2014! \mod(2017)=\boxed{1008}

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice Answer!

Victor Paes Plinio - 6 years, 11 months ago

same method!

Adarsh Kumar - 6 years, 10 months ago

Wilson's theorem

Krishna Shankar - 6 years ago
Ilham Akbar
Mar 24, 2015

If 2! Mod 5 = 2 ,4! Mod 7 = 3 , 8! Mod 11 = 5 , and 10 ! Mod 13 = 6 so this is an formula for prime mod and the remainder is P-1 /2 so 2014! Mod 2017 = 2017-1/2 = 1008.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...