Divisibility test using 9 and 11 (for 99) Which of the following integers is exactly divisible by 99 (i.e., divisible by both 9 and 11)?

Introduction / Context:
To be divisible by 99, a number must be divisible by both 9 and 11. This question tests your fluency with quick divisibility checks without doing long division.

Given Data / Assumptions:

• Four candidate numbers are provided.
• Rule for 9: sum of digits must be a multiple of 9.
• Rule for 11: (sum of digits in alternating positions) must differ by a multiple of 11 (including 0).

Concept / Approach:
Apply the tests sequentially. First screen with the easy 9-test (digit sum). For those passing, check the 11-test using alternating sum starting from the leftmost digit with +, −, +, −, … signs.

Step-by-Step Solution:
114345: digit sum = 1+1+4+3+4+5 = 18 → multiple of 9.11-test: 1−1+4−3+4−5 = 0 → multiple of 11 → passes.Other options fail at least one test: e.g., 135792 has digit sum 27 (ok) but 11-test gives 3 (not multiple of 11); 913464: 27 (ok) but 11-test gives 9; 3572404 digit sum 25 (fails 9-test).

Verification / Alternative check:
Since 114345 passes both 9 and 11 tests, it is divisible by 99 (because 9 and 11 are coprime).

Why Other Options Are Wrong:
3572404: fails divisibility by 9.135792 and 913464: pass 9 but fail the 11-test.

Common Pitfalls:
Mixing the 11-test sign pattern or summing all digits for the 11-test instead of alternating signs. Also, stopping after the 9-test and forgetting to check 11.

Final Answer:
114345