Prime Numbers and Divisibility

**Aryth** · April 9th 2009, 12:26 PM

What is the remainder when 5!25! is divided by 31?

**CaptainBlack** · April 10th 2009, 11:21 AM

Originally Posted by Aryth

What is the remainder when 5!25! is divided by 31?

$5! 25! \equiv 5! 23! (-7) (-6) \equiv 5! 23! 11 \mod 31$

................. $\equiv 5! 22! 5 \mod 31$

................. $\equiv 5! 21! 17 \mod 31$

etc

CB

**JaneBennet** · April 10th 2009, 12:36 PM

Or use Wilson’s theorem.

$30!\equiv-1\pmod{31}$

$30\times29\times28\times27\times26\equiv(-1)(-2)(-3)(-4)(-5)\pmod{31}\equiv-5!\pmod{31}$

$\therefore\ 5!25!\equiv-30!\pmod{31}\equiv1\pmod{31}$

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