Which of the following numbers is divisible by 99?

## Concept & Strategy To determine if a large number is divisible by $99$, check if it is divisible by its co-prime factors: **$9$ and $11$**. * **Rule of 9:** The sum of all digits must be a multiple of 9. * **Rule of 11:** The difference between the sum of digits in alternating positions must be 0 or a multiple of 11. Always test the rule of 9 first, as it is computationally faster and often eliminates options instantly. ## Step-by-Step Solution **Calculation / Deduction:** * **Step 1: Test all options using the Rule of 9.** * *(a) 114345:* Sum $= 1 + 1 + 4 + 3 + 4 + 5 = 18$. (Divisible by 9, keep) * *(b) 913464:* Sum $= 9 + 1 + 3 + 4 + 6 + 4 = 27$. (Divisible by 9, keep) * *(c) 135792:* Sum $= 1 + 3 + 5 + 7 + 9 + 2 = 27$. (Divisible by 9, keep) * *(d) 3572404:* Sum $= 3 + 5 + 7 + 2 + 4 + 0 + 4 = 25$. (Reject) * **Step 2: Test the remaining options using the Rule of 11.** * *(a) 114345:* Odd positions (from right): $5 + 3 + 1 = 9$ Even positions (from right): $4 + 4 + 1 = 9$ Difference: $9 - 9 = 0$. (Divisible by 11. **Accept!**) * *(b) 913464:* Odd positions (from right): $4 + 4 + 1 = 9$ Even positions (from right): $6 + 3 + 9 = 18$ Difference: $9 - 18 = -9$. (Reject) * *(c) 135792:* Odd positions (from right): $2 + 7 + 3 = 12$ Even positions (from right): $9 + 5 + 1 = 15$ Difference: $12 - 15 = -3$. (Reject) ## Exam Strategy & Shortcut For the rule of 9, use the **Casting Out Nines** technique. Instead of adding every single number, cross out any individual $9$s or any groups of numbers that sum to $9$. * For (a) 114345: $4+5=9$ (cross out). Remaining: $1+1+4+3 = 9$ (cross out). Remainder is 0, so it passes. This saves mental energy for the rule of 11, which requires more careful tracking of alternating digits. ## Common Pitfall The biggest time-waster here is starting with the rule of 11. While robust, writing out the odd/even sums for four different six/seven-digit numbers takes considerably longer than a rapid mental digit sum. Always sequence your divisibility tests from the simplest (2, 5, 10), to moderate (3, 9, 4), to complex (7, 11, 13). ## Final Answer **Therefore, the correct answer is 114345.**