# Wilson's Theorem

## Contents

## Statement

If and only if is a prime, then is a multiple of . In other words .

## Proof

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 a 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 1 and , then the others can be grouped into pairs whose product is congruent to one,

Finally, multiply this equality by to complete the proof.

## Example

Let be a prime number such that dividing by 4 leaves the remainder 1. Show that there is an integer such that is divisible by .

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