Question

Show that if n is a composite integer with \(n\neq 4\) , then \((n-1)! \equiv 0~(mod n\)
Please, help

Answer 1

" n is a composite integer" the next largest after n = 4 is n = 6

if n = 6, then (n-1)! = (6-1)! = 5! = 5*4*3*2*1

Notice how 2*3 = 6 is a factor, so 5! = 0 (mod 6)

If n = 8 (the next composite integer) then

(n-1)! = (8-1)! = 7! = 7*6*5*4*3*2*1

and we see 4*2 = 8 as a factor, so 7! = 0 (mod 8)

-------------------------------

In general, if n is composite then n = p*q where p and q are smaller integers that will be factors of (n-1)! since the factorial counts down to 1