If and only if p is a prime, then is a multiple of p. Written more mathematically,
Wilson's theorem is easily verifiable for 2 and 3, so let's consider . If is composite, then its positive factors are among Hence, , so .
However if is prime, then each of the above integers are relatively prime to . So for each of these integers there is another such that . It is important to note that this is unique modulo , and that since is prime, if and only if is or . Now if we omit and , then the others can be grouped into pairs whose product is congruent to one,
Finally, multiply this equality by p-1 to complete the proof. Insert non-formatted text here