Which of the following numbers is exactly divisible by 24?

## Concept & Logic To check for divisibility by a composite number like 24, we must break it down into its **co-prime factors**. The co-prime factors of $24$ are $8$ and $3$ (since $8 \times 3 = 24$ and they share no common factors other than 1). Therefore, a number is divisible by 24 if and only if it satisfies **both** the rule for 8 and the rule for 3. * **Divisibility by 8:** The last three digits of the number must be divisible by 8. * **Divisibility by 3:** The sum of all digits must be divisible by 3. ## Step-by-Step Solution **Calculation / Deduction:** Always test the stricter rule first to eliminate options quickly. Testing divisibility by 8 on the last three digits is much faster than summing up a 7-digit number. * **Step 1: Check rule of 8 (Last three digits)** * *(a) 35718:* Last three digits are 718. $718 \div 8 = 89$ remainder 6. (Reject) * *(b) 63810:* Last three digits are 810. $810 \div 8 = 101$ remainder 2. (Reject) * *(c) 537804:* Last three digits are 804. $804 \div 8 = 100$ remainder 4. (Reject) * *(d) 3125736:* Last three digits are 736. $736 \div 8 = 92$ exactly. **(Accept)** * **Step 2: Check rule of 3 for the remaining candidate (d)** * Number: 3125736 * Sum of digits: $3 + 1 + 2 + 5 + 7 + 3 + 6 = 27$ * Since 27 is a multiple of 3, the number is divisible by 3. Since 3125736 is divisible by both 8 and 3, it is exactly divisible by 24. ## Exam Strategy & Shortcut **Rule Stacking:** Whenever you have a composite divisibility question, never test the addition rule (like 3 or 9) first. Addition rules require processing the entire number. Always test the "tail" rules first (like 2, 4, 8, 5, 25). By isolating the last three digits (718, 810, 804, 736) and testing for 8, we eliminated three options in seconds without doing a single addition. ## Common Pitfall A fatal mistake is breaking 24 down into factors that are NOT co-prime, such as 12 and 2, or 6 and 4. If you check for divisibility by 6 and 4, you might get false positives. For example, the number 12 is divisible by 6 and 4, but it is certainly not divisible by 24! Always use co-prime factors (8 and 3). ## Final Answer **Therefore, the correct answer is 3125736.**