Which of the following are divisible by 2, 3 and 11? A. 8448 B. 9812 C. 9126…

A number is divisible by 66 (= 2 x 3 x 11) exactly when it independently satisfies the tests for 2, 3 and 11: even last digit (test for 2), digit sum a multiple of 3 (test for 3), and alternating digit sum -- sum of digits at odd places from the right minus sum of digits at even places -- equal to 0 or a multiple of 11 (test for 11).

1. 8448: last digit 8 is even; digit sum 8+4+4+8 = 24, a multiple of 3; alternating sum 8-4+4-8 = 0. All three tests are satisfied.

2. 9812: last digit 2 is even; digit sum 9+8+1+2 = 20, not a multiple of 3; alternating sum 2-1+8-9 = 0. The divisibility-by-3 test fails.

3. 9126: last digit 6 is even; digit sum 9+1+2+6 = 18, a multiple of 3; alternating sum 6-2+1-9 = -4, not a multiple of 11. The divisibility-by-11 test fails.

4. 9636: last digit 6 is even; digit sum 9+6+3+6 = 24, a multiple of 3; alternating sum 6-3+6-9 = 0. All three tests are satisfied.

Cross-check by dividing directly by 66: 8448 / 66 = 128 exactly and 9636 / 66 = 146 exactly, confirming both are multiples of 66; 9812 / 66 is approximately 148.67 and 9126 / 66 is approximately 138.27, neither a whole number, confirming neither is a multiple of 66.

So the numbers divisible by 2, 3 and 11 are 8448 and 9636.