so I dont know if this is really what the website is meant for so let me know. I have a midterm on abstract algebra (intro) and I've been doing practice problems and I just need to know whether my answers are correct and if my way of doing them is right.
1) Does the following have a solution: $x^2 \equiv 1 \pmod{3}$ Well we know that a solution exists if $\mathrm{gcd}(a, n) | b$ where $a = x^2$, $b = 1$ and $n = 3$. we can rewrite the equation as $1 \equiv x^2 \pmod{3}$
2) If $a \equiv 3 \pmod{4}$ prove that there are no integers c, d such that $a = c^2 + d^2$ By definition we can write this as $a = 4n + 3, n \in \mathbb{Z}$. so does this prove it? since $4n + 3$ can never be in the form of $c^2 + d^2$.
3) Show that no perfect square has 2, 3, 7, 8 as its last digit. The way we did this was we listed out all the classes from in $\mathbb{Z}_{10}$ which are $[0]_{10}..[9]_{10}$ and showed that $a^2 (0, 1, 4, 16, 25 ...)$ are never an element of the $[2], [3], [7], [8]$
4) If $a \in \mathbb{Z}$, prove that last digit of $a^4$ is $0, 1, 5, \text{or } 6$