Problem 5. A multiple-choice test consists of 20 questions. For each correctly solved problem, 8 points are awarded, and for each incorrectly solved problem, 5 points are deducted. If no answer is circled for a question, 0 points are given for it. A student ends up with 13 points. How many problems did the student solve correctly?
Solution. Let $x$ be the number of correctly solved problems, $y$ be the number of incorrectly solved problems, and $z$ be the number of problems on which the student did not answer at all. According to the conditions of the problem, we obtain the system

$$
\left\{\begin{array}{l}
x+y+z=20 \\
8 x-5 y=13
\end{array}\right.
$$

where $x, y$, and $z$ are non-negative integers.

From the second equation, we get

$$
\begin{aligned}
& 8 x=13+5 y \\
& x=\frac{13+5 y}{8}=1+\frac{5+5 y}{8}=1+\frac{5}{8}(1+y)
\end{aligned}
$$

from which, for $x$ to be an integer, $1+y$ must be divisible by 8, i.e., $1+y=0$ or $1+y=8$ or $1+y=16$. In this case,

1) $y=-1$, which is not possible, since $y$ cannot be negative.
2) $y=7, x=\frac{13+5 y}{8}=\frac{48}{8}=6, z=20-6-7=7$
3) $y=15, x=\frac{13+5 \cdot 15}{8}=11, z=20-15-11=-6<0$, which is not possible since $z \in \mathbb{N}$.

Therefore, the student answered correctly to 7 problems.