## 13. Caramels

Martina distributed 26 caramels to her four little brothers. Each of the brothers ate a few caramels, and an hour later, Martina found that each of them had the same number of caramels left. Suppose the oldest brother ate as many as the third, the second ate half of his caramels, and the fourth ate as many as the other three.

How many caramels did Martina give to each of the brothers?
13. Let $x$ be the number of candies left with each of the brothers, and $y$ be the number of candies eaten by the elder (or third) brother. Then the second brother ate $x$ candies, and the fourth brother ate $-(y+x+y)$ candies.

In total, the number of candies eaten is

$$
y+x+y+(x+2 y)=2 x+4 y
$$

candies. On the other hand, we know that $4 x$ candies are left. Therefore,

$$
4 x+(2 x+4 y)=26
$$

or

$$
6 x+4 y=26, \quad 3 x+2 y=13
$$

But $x$ and $y$ are positive integers, different from 1. Clearly, $3 x=13-2 y$ is odd; hence, $x$ is odd, and $x<4$ (since $3 x<13$). Therefore,

$$
x=3 \quad \text { and } \quad y=2
$$

Thus, the elder brother received 5 candies, the second brother - 6 candies, the third - 5, and the fourth - 10 candies.