Please see attached document.
There are four problems. The first problem is
71! mod 73
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.
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
This equation has no solution. If it has a solution , then . We note , then we get . Since , then . Thus ...
This shows how to find the least non-negative residue and solve a congruence.