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 .
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.