1. Determine the smallest natural number that is divisible by 225 and whose decimal representation consists only of the digits 0 and 1.
Solution. Since $225=9 \cdot 25$, it follows that the desired number is divisible by 9 and by 25. Since the digits used to write the number are only 0 and 1, for the number to be divisible by 25, it needs to end in two zeros, and for it to be divisible by 9, the sum of its digits must be divisible by 9. We are looking for the smallest number, so from the divisibility by 9, it follows that the number must have nine ones in its representation. Therefore, the desired number is 11111111100.