5. In a box, there are 67 balls of two sizes (small and large) and two colors (white and red). It is known:

- the number of red balls is divisible by 5;
- the number of large red balls is equal to the number of white balls;
- of the four types of balls, the smallest number is small white;
- the number of balls of each type is prime.

Determine how many balls of each type are in the box.

## Fourth grade - B category
5. Let $C$ be the number of large red balls, $c$ the number of small red balls, $B$ the number of large white balls, and $b$ the number of small white balls. Then all these numbers are prime and $C+c+B+b=67$. For the sum of four prime numbers to be odd, one of them must be 2 (since the sum of four odd numbers is even). Since 2 is the smallest prime number, it follows from the third condition that $b=2$. This leaves $C+c+B=65$. Since $5 \mid C+c$ (the first condition), it follows that $5 \mid B$, so $B=5$ (since $B$ is a prime number). Now, from the second condition, we get $C=B+b=7$, and then $c=53$ remains. Therefore, the balls in the box are: 7 large red, 53 small red, 5 large white, and 2 small white.

## Fourth Grade - Category B