Find the smallest number n for which 11 divides 20n^(2)-1.

For this question, I have made it so that 20n^2 is congruent to -1mod11; the quadratic residues of 11 are

1 goes to 1, 2 goes to 4, 3 goes to 9, 4 goes to 5, 5 goes to 3. But none of these contain the number -1, so -1 does not belong in the set of squares, so how am I supposed to work out this question?

I must have done something wrong in my understanding, so could anyone explain to me what I have done wrong?

Thanks!

For this question, I have made it so that 20n^2 is congruent to -1mod11; the quadratic residues of 11 are

1 goes to 1, 2 goes to 4, 3 goes to 9, 4 goes to 5, 5 goes to 3. But none of these contain the number -1, so -1 does not belong in the set of squares, so how am I supposed to work out this question?

I must have done something wrong in my understanding, so could anyone explain to me what I have done wrong?

Thanks!

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