2018/19 AHSMC Part 1 Problem 6

Let $x$ be the number of correct answers and $y$ the number of non-answered questions. Thus, $0 \le x + y \le 16$ and $$20 + 5x + 2y = 78 \iff 5x + 2y = 58.$$ Since $5x = 2(29 - y)$, $5$ should be a divisor of $29 - y$, hence $y \in \{4, 9, 14\}$. The only convenient value for $y$ is $4$ and hence the solution is $x = 10$, $y = 4$. This gives $10$ correct answers, $2$ wrong answers and $4$ no answers. The answer is (C).

Alternative solution: The total for a perfect paper is $100$. If the student didn't answer $a$ questions and the answers for $b$ questions were not correct, then $$3a + 5b = 22$$ where $0 \le a + b \le 16$. It is clear that $b \le 4$ and since $5b \equiv 1 \pmod{3}$ one obtains $b = 2$ and then $a = 4$. Therefore, the remaining $10$ questions are correct.