10.1. The student was given an assignment consisting of 20 problems. For each correctly solved problem, they receive 8 points, for each incorrectly solved problem - minus 5 points, and for a problem they did not attempt - 0 points. The student scored a total of 13 points. How many problems did the student attempt?
# Solution.

Let $x$ be the number of problems solved correctly, $y$ be the number of problems solved incorrectly. Then we get the equation $8x - 5y = 13$. This equation can be solved in two ways.

1st method. Rewrite the equation as $8(x + y) = 13(1 + y)$. We see that the number $x + y$ is divisible by 13. On the other hand, by the condition, $x + y$ is no more than 20. Therefore, $x + y = 13$ (in this case, $x = 6$, $y = 7$).

One solution to the equation is immediately guessed: $x_0 = 1, y_0 = -1$. For any integer $t$, the pair of numbers $x = 1 + 5t, y = -1 + 8t$ also satisfies this equation.

Answer: 13 problems.

Other methods of solution are possible.

Criteria: only one correct answer is given without justification - 1 point.