\begin{align*} \text{Question 1:} &\quad 1039x + 2013y = 4 \iff 1039x = 4-2013y \iff 1039x \equiv 4 \mod 2013 \iff x = 1039^{-1} 4 \mod 2013 \\
\text{Question 4}: & \quad 3 \mathrel{|} (x^2 + y^2) \implies \exists k \in \mathbb Z \quad s.t. \quad x^2 + y^2 = 3k \implies 3 \mathrel{|}x^2, y^2...