6. Find the smallest natural number $N$ such that the number $99 N$ consists only of threes.

ANSWER: 3367.
Solution. The number 33N must consist of all ones. A number is divisible by 33 if it is divisible by 3 and by 11. A number consisting of all ones is divisible by 3 if the number of ones is a multiple of 3, and it is divisible by 11 if the number of ones is a multiple of 2. The smallest such number is 111111, so 33N = 111111, from which \( N=3367 \).

Lomonosov Moscow State University

## School Olympiad "Conquer Sparrow Hills" in Mathematics

Final Round Tasks for 2015/2016 Academic Year for 7-8 Grades