Application of Wilson's Theorem

NuncChaos · Poincare Conjecture Offline Joined: 03 Feb 2008 Posts: 140

Given that $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} +.... + \frac{1}{23} = \frac{n}{23!}$

Find the remained when $n$ is divided by 13.

the answer is

My question is, how exactly does one apply Wilson's to this problem?
And is there a more elementary method?

**Posted:** Sat Oct 31, 2009 9:46 pm

zapi2007

here we go

wondering if there's an easier way to clean up steps 6, 7, and 8... maybe

for an elementary method, well, the first few steps are straightforward. But, it gets messy when you need to find $\frac{23!}{13} mod 13$ Sad

Brute force? The most elementary thing there is. Very Happy

But if you didn't know Wilson's Theorem, I'd bash 10! twice, then multiply by 11 and 12, knowing that 11=13-2 and 12=13-1. You can also do it with 9! if you feel like you need to make it more balanced between finding the factorial and long dividing, or multiplying it later with mod 13 in mind.

**Posted:** Sat Oct 31, 2009 10:16 pm

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JAM · Riemann Hypothesis Offline Joined: 14 Jun 2004 Posts: 495

**Posted:** Sun Nov 01, 2009 9:22 am

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