Task B-3.3. The number $n$ is the smallest positive multiple of 15 whose digits are all 0 or 8. Determine the largest prime number by which the number $n$ is divisible.
## Solution.

The number $n$ must be divisible by 5, so its last digit is 0.

(1 point)

The number $n$ must be divisible by 3, so the sum of its digits must be divisible by 3.

Since its only digits are 0 or 8, the sum of the digits is a multiple of 8, and the smallest multiple of 8 that is divisible by 3 is 24.

(2 points)

We conclude that the desired number must have three digits 8 and the last digit 0, so 8880 is the desired number.

(1 point)

Its prime factorization is $8880=2^{4} \cdot 3 \cdot 5 \cdot 37$. Therefore, the largest prime divisor of 8880 is 37.

(2 points)