Find all quadratic residues modulo $15$.

Question

Find all quadratic residues modulo $15$.

I'm having trouble understanding how my professor got the solutions.

I know that the quadratic residue is $x^2 \equiv q \bmod p.$

The answers she got was $1,4,6,9,10.$

Since

$1^2 \equiv 1 (\bmod15)$

$4^2 \equiv 1 (\bmod15)$

$6^2 \equiv 2 (\bmod15)$

$9^2 \equiv 6 (\bmod15)$

$10^2 \equiv 10 (\bmod15)$

I'm trying to grasp on how she got the answers, and why $ 5^2$ or $7^2$ won't work?

Edit: Thank you for the quick responses, I understand now how they got the quadratic residues, it ended up being easier after seeing the answers Thank you!

Answer 1

Recall that $a$ is a quadratic residue modulo $p$ if there exists $x\in\Bbb Z$ such that $x^2\equiv a\bmod p$ has a solution.

So, working in modulo $15$; we have $$1^2=1,\quad 2^2=4,\quad 3^2=9,\quad4^2=1,\quad 5^2=10,\quad6^2=6,\quad7^2=4,$$ $$8^2=4,\quad 9^2=6,\quad 10^2=10,\quad 11^2=1,\quad 12^2=9,\quad 13^2=4,\quad14^2=1.$$

So the quadratic residues modulo $15$ are $\{1,4,9,10,6\}.$

Answer 2

You have the definition a bit backward. $1^2 \equiv 1,$ so $1$ is a quadratic residue, $2^2 \equiv 4,$ so $4$ is a quadratic residue, and so on. The quadratic residues are the things which are themselves squares mod $p;$ you shouldn't attempt to square them.