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hello can someone please help me to solve this problem:

2008 mod 71,

9 square mod 41,

34 suare mod 71

b)determine all a and b that verify

a square mod 41=40

b square mod 71=20

this is my answer:

a) 2008 mod 41=40

2008 mod 71=20

9 square mod 41=40

34 suare mod 71=20

b) i noticed that a=9 is a solution for : a square mod 41=40

and b=34 is a solution for : b square mod 71=20

but the problem require finding all the solution .can some one please help me .

1 Answers1

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As $41$ is a prime number, $\mathbf Z/41\mathbf Z$ is a field/ In any field, a quadratic polynomial has at most 2 roots. In particular, an element has at most 2 roots, which are opposite. The polynomial $x^2-40$ has $9$ as a root, hence $-9=32$ is the other root.

Same argument for $x^2\equiv 20\mod 71$ since $71$ is prime.

Bernard
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