This

neat little

theorem is sometimes useful (especially in various

Mathematics exams!).

**Theorem.** Let *p* be a prime number. Then (*p*-1)! = -1 (modulo *p*).

The

proof usually takes one of many similar forms; it's quite

simple, but surprising the first few times you see it.

"Recall" that the integers modulo some prime number *p* form a field. If *p*=2, the theorem is true ((1-1)! = 1 = -1 modulo 2).

If *p* is an odd prime, then 1 != -1 (modulo *p*). Thus these are the only two solutions of the equation *x*^{2}=1 modulo *p*. (*p*-1)! modulo *p* is the product of all non-zero elements of the integers modulo *p*. So every element *a* apart from 1 and -1 is also multiplied by its inverse; the product is 1. Thus the product (modulo *p*) is exactly -1 (since -1 is *not* multiplied by its inverse!). QED