Prove that if $n \in \mathbb{Z}$ such that $n^3$ is a perfect square, then $n$ is a perfect square.

Find the following, showing all of your computations and explanations:

• all integers in $(\mathbb{Z} / 7 \mathbb{Z})^{\times}$that are not quadratic residues modulo 7
• the number of integers between 1 and 76 that are relatively prime to 76
• a positive integer that is three more than a multiple of 11 and whose last two digits are 32 .

Find the six values of $m$ for which 3 has order 4 modulo $m$. That is, find the moduli $m$ for which $\operatorname{ord}_m(3)=4$.

Find two solutions to the following quadratic congruence:
$$
x^2+7 x+11 \equiv 0 \quad \bmod 139 .
$$
If this congruence has no solutions, explain how you know. You may not use any calculators for this question.