How to Prove that (n-1)!= 0 (mod n) for Composite n?

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This discussion focuses on proving that for any composite integer n greater than 4, (n-1)! is congruent to 0 modulo n. The proof utilizes the factorization of n into two distinct integers, i and j, where n = (n-i)(n-j). The key insight is that if i is not equal to j, both factors will appear in the expansion of (n-1)!, thereby ensuring that (n-1)! is divisible by n.

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dessy
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If n is composite, n>4, prove that (n-1)!= 0 (mod n). = is congruent
 
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Let n=(n-i)(n-j). If i is not equal to j then both terms will appear in the expansion of (n-1)!. You can probably handle the case i=j yourself with a little thought.
 

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