How many of the following numbers are divisible by 3 but not by 9? 2133, 2343, 3474, 4131, 5286, 5340, 6336, 7347, 8115, 9276

## Concept & Logic This problem tests the relationship between the divisibility rules of 3 and 9. * **Divisibility by 3:** The sum of the digits must be a multiple of 3. * **Divisibility by 9:** The sum of the digits must be a multiple of 9. Every number divisible by 9 is also divisible by 3, but the reverse is not true. We need to find numbers whose digit sum is a multiple of 3, but strictly **not** a multiple of 9. ## Step-by-Step Solution **Calculation / Deduction:** Let's calculate the sum of the digits for each number and evaluate both conditions: * **2133:** $2 + 1 + 3 + 3 = 9$. Divisible by 9. (Reject) * **2343:** $2 + 3 + 4 + 3 = 12$. Divisible by 3, NOT 9. **(Accept)** * **3474:** $3 + 4 + 7 + 4 = 18$. Divisible by 9. (Reject) * **4131:** $4 + 1 + 3 + 1 = 9$. Divisible by 9. (Reject) * **5286:** $5 + 2 + 8 + 6 = 21$. Divisible by 3, NOT 9. **(Accept)** * **5340:** $5 + 3 + 4 + 0 = 12$. Divisible by 3, NOT 9. **(Accept)** * **6336:** $6 + 3 + 3 + 6 = 18$. Divisible by 9. (Reject) * **7347:** $7 + 3 + 4 + 7 = 21$. Divisible by 3, NOT 9. **(Accept)** * **8115:** $8 + 1 + 1 + 5 = 15$. Divisible by 3, NOT 9. **(Accept)** * **9276:** $9 + 2 + 7 + 6 = 24$. Divisible by 3, NOT 9. **(Accept)** Counting the accepted numbers, we have exactly 6 numbers that meet the criteria. ## Exam Strategy & Shortcut Instead of writing out the full sum, use the **Digit Cancellation Method (Casting out 9s)**. Cross out any digits (or combinations of digits) that sum to 9. If the remaining digits sum to 3 or 6, the number is divisible by 3 but not 9. If the remaining sum is 0 (or 9), it is divisible by 9. * *Example 6336:* Cross out 6 and 3 (sums to 9), cross out 3 and 6 (sums to 9). Remainder is 0. Divisible by 9. Skip it. * *Example 7347:* 7+2=9 (borrow 2 from 3, leaves 1). 1+4+4(from 7)=9. Remaining is 3. Divisible by 3, not 9. Keep it. ## Common Pitfall A major trap is calculating the sum of digits, confirming it's divisible by 3, and immediately counting it without checking if it's also a multiple of 9. The question specifically excludes multiples of 9, so filtering them out is the core task. ## Final Answer **Therefore, the correct answer is 6.**