LogFAQs > #978889442

willythemailboy posted...

n is not prime and n is greater than 4.

Yeah this is correct. n = 6, 8, 9, 10, 12, 14, 15, 16, 18...etc divide (n-1)!

Here's the why part. For n<5 just calculate by hand so here's the proof for n>=5
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(1) If n is not a prime then by definition it could be expressed as n = x*y with 1<x<n and 1<y<n. (n-1)! = (n-1)*(n-2)...*2*1 will automatically have these x and y as factors. Since x and y are both factors of (n-1)!, x*y = n will also be a factor of (n-1)!

In case of n = x*x instead of n = x*y (example 49 = 7*7), (n-1)! will have those two x as factors from x itself (7) and 2x (14=2*7, got another 7 from here). The rest is the same as above.

(2) If n is a prime then according to Wilson's theorem n divides (n-1)!+1. However this would contradict the problem in which n divides (n-1)! and not (n-1)!+1. Therefore n cannot be a prime.
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Proof(2) is proof by contradiction. I could ELI5 that proof(2) but I'm too lazy for that and also...
EPR-radar posted...

if TC is getting his math homework done online, the exams are gong to be ugly.