## Task 4 - 170834

A pioneer group collected waste paper; the entire proceeds were transferred to the solidarity account. The pioneers formed two brigades, with each pioneer in the group belonging to exactly one of these brigades. The following is known about the collection results:

(1) Each pioneer in Brigade $A$ collected exactly $13 \mathrm{~kg}$, except for one who only brought $6 \mathrm{~kg}$.

(2) Each pioneer in Brigade $B$ collected exactly $10 \mathrm{~kg}$, except for one who only brought exactly $5 \mathrm{~kg}$.

(3) Brigade $A$ collected the same amount in total as Brigade $B$.

(4) The entire pioneer group collected more than $100 \mathrm{~kg}$ but less than $600 \mathrm{~kg}$ of waste paper.

a) How many pioneers belonged to each brigade in total?

b) How many Marks could the pioneer group transfer to the solidarity account if the scrap dealer paid 0.15 Marks per kg of waste paper?
a) Let the number of pioneers who each collected exactly $13 \mathrm{~kg}$ be $x$. According to (1), then exactly $(x+1)$ pioneers belong to Brigade A, and the collection result of this brigade was $(13 x+6) \mathrm{kg}$.

The number of pioneers who each collected exactly $10 \mathrm{~kg}$ be $y$. According to (2), then exactly $(y+1)$ pioneers belong to Brigade B, and the collection result of this brigade was $(10 y+5) \mathrm{kg}$.

According to (3), we have $13 x+6=10 y+5$, thus $y=\frac{13 x+1}{10} \quad(*)$.

$13 x+1$ is divisible by 10, so the (digit representation of) number $13 x$ ends with the digit 9, and thus $x$ ends with the digit 3.

From (4) it follows that $100<2(13 x+6)<600$ and thus $44<13 x<294$.

This is not satisfied by $x=3$ nor by the numbers $x=23$, as for these $13 x=39$ and $13 x=299$ respectively. Therefore, $x=13$.

From $\left({ }^{*}\right)$, it follows that $y=17$. Thus, exactly 14 students belonged to Brigade A and exactly 18 to Brigade B.

b) Brigade A collected $(13 \cdot 13+6) \mathrm{kg}=175 \mathrm{~kg}$ and Brigade B $(17 \cdot 10+5) \mathrm{kg}=175 \mathrm{~kg}$, so the entire group collected $350 \mathrm{~kg}$ of waste paper.

Since $350 \cdot 0.15 \mathrm{M}=52.50 \mathrm{M}$, the pioneer group could transfer exactly $52.50 \mathrm{M}$ to the solidarity account.