5. In the $3000-$th year, the World Hockey Championship will be held under new rules: 12 points will be awarded for a win, 5 points will be deducted for a loss, and no points will be awarded for a draw. If the Brazilian national team plays 38 matches in this championship, scores 60 points, and loses at least once, how many victories can they achieve? Provide all possible options and explain why there cannot be any others.
Solution. Let Brazil win in x matches and lose in y matches. We form the equation $12 x-5 y=60$. We see that $12 \mathrm{x} \vdots 12$ and $60 \vdots 12$. GCD(5, 12)=1, i.e., $y \vdots 12$. Possible: a) $y=12$. Then we get the equation $12 x-60=60$. Thus, $x=10$. This is possible. b) $y=24$. We get the equation: $12 x-120=60$. Hence, $x=15$. This is impossible, as the number of matches already exceeds 38. c) $y=36$. We get $12 x-180=60$. Hence, $x=20$, which is also impossible. Larger values of x do not fit, as the number of matches played will already exceed 38.

Answer. 10 wins.

Recommendations for checking. Only the correct answer - 0 points. Only the correct answer with verification - 1 point. The equation is correctly set up, but there is no further progress (even with the correct answer and verification) - 2 points. The equation is set up, there is an idea of divisibility, but there is no further progress - 3 points. After setting up the equation, there is an idea of divisibility, only the first case is considered, and the correct answer is obtained - 4 points. After setting up the equation, there is an idea of divisibility, only the first and second cases are considered, and the correct answer is obtained - 5 points. After considering the second case without explanation, it is written that the number of matches increases further - do not deduct points. Full enumeration with one computational error and the correct answer - 5 points. One case is missed in the full enumeration - no more than 4 points. More than one case is missed in the full enumeration with the correct answer - 1 point, with the wrong answer - 0 points.