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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.
    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 ...

    Solution Summary

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