# Theory of numbers

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Use Quadratic Congruences to evaluate the following fractions. See attached file for full problem description.

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##### Solution Summary

The solution involves the result concerning quadratic congruences.

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Euler's Criterion

Let p be an odd prime and let a be an integer not divisible by p. If a is a square modulo p, then . If a is not a square modulo p, then

Quadratic Residues:

Let p be a prime, and let a be an integer not divisible by p. If a is a square modulo p, then a is said to be a quadratic residue modulo p. Otherwise, we say that a is a quadratic nonresidue modulo p.

Legendre symbol:

Let p be a prime, and let a be an integer. The Legendre symbol is defined by

Theorem ...

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