Find the number of digits in the smallest number which is made up of digits 1 and 0 only and is divisible by 225.

### Concept & Logic When a number is divided by a composite number like 225, break it down into its co-prime factors. $$ 225 = 9 \times 25 $$ A number is divisible by 225 if it is independently divisible by both 9 and 25. - Divisibility by 25: The last two digits must be 00, 25, 50, or 75. - Divisibility by 9: The sum of all its digits must be a multiple of 9. ### Step-by-Step Solution - **Given:** The smallest number made strictly of the digits 1 and 0, which is divisible by 225. - **Calculation (Divisibility by 25):** Since the number can only contain 1s and 0s, the last two digits must be 00 to satisfy the divisibility rule for 25. - **Calculation (Divisibility by 9):** The sum of the digits must be a multiple of 9. Since the available digits are only 1 and 0, the smallest way to get a digit sum of 9 is to use exactly nine 1s. - **Deduction:** Combining these two conditions, the smallest number consists of nine 1s followed by two 0s to satisfy both rules simultaneously. This number is 11111111100. - Total digits = 9 (from the 1s) + 2 (from the 0s) = 11 digits. ### Exam Strategy & Shortcut Treat the conditions modularly. The "00" at the end handles the 25 divisibility and contributes zero to the digit sum. Thus, you only need to count how many 1s are needed to hit a sum of 9. Nine 1s plus two 0s equals 11 digits. No complex division is required. ### Common Pitfall A major mistake is finding the correct number (11111111100) but then forgetting to answer the specific question asked, which is the *number of digits*, not the number itself. Always re-read the final sentence of the prompt before selecting an option. ### Final Answer Therefore, the correct answer is 11.