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# Number theory

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There are four problems. The first problem is
71! mod 73

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#### 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.

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