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Lagrange's theorem

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Suppose that...

Use Lagrange's Theorem

Suppose that a N and

a z (mod pq) where p q 3(mod 4 ) are primes.

Prove that there are only four possible square roots of a modulo pq, and they are given as follows. For x,y Z given by the extended Euclidean Algorithm, such that xp +yq =1

We have
Z= (xpa + yqa ), and Z= (xpa - yqa )

Use Lagrange's Theorem

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Solution Summary

This is a proof involving Lagrange's theorem.

Solution Preview

Please see the attachment.

Proof:
The equation (mod ) is equivalent to the system of equations
(mod ) and (mod )
Now let's consider the equation (mod ). It has two solutions if is a quadratic residue module . Since ...

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