# Number theory

Please see attached document.

There are four problems. The first problem is

71! mod 73

https://brainmass.com/math/number-theory/upper-level-number-theory-problems-27910

#### Solution Preview

Please see the attachment.

The notation means divides or is a multiple of .

1. Find the least non-negative residue.

In this problem, we should use the Wilson's Theorem: ( ) for any prime number .

We know is a prime number, according to the Wilson's Theorem, we have

( ) (1)

But we note ( ), so ( )

We multiply on both sides for equation (1), then we get

( )

Thus the least non-negative residue of is .

2. Solve the congruence.

(a)

We note , we need to find . We use the following procedure.

Thus we have

So we get . This means is . Thus we have

Therefore, the solution of the congruence is

(b)

This equation has no solution. If it has a solution , then . We note , then we get . Since , then . Thus ...

#### Solution Summary

This shows how to find the least non-negative residue and solve a congruence.