Task 1. A group of boys and girls collected 1700 denars for a birthday gift. The girls collected 200 denars each, and the boys collected 300 denars each. Determine the number of boys and girls in the group, given that the group has an odd number of members.
Solution. If we denote the number of girls by $x$, and the number of boys by $y$, then we have $200 x + 300 y = 1700$, i.e., $2 x + 3 y = 17$. This means the number $2 x + 3 y$ must be odd, so $y$ must be an odd number and $y \in \{1, 3, 5\}$.

If $y = 1$, then $x = \frac{17 - 3}{2} = 7$, but then the group has $1 + 7 = 8$ members, which is not possible.

If $y = 3$, then $x = \frac{17 - 9}{2} = 4$ and the group has $3 + 4 = 7$ members.

If $y = 5$, then $x = \frac{17 - 15}{2} = 1$, but then the group has $5 + 1 = 6$ members, which is not possible.

Finally, the group has 4 girls and 3 boys.