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The digits indicated by * in 3422213** so that this number is divisible by 99 areOptions1,94,65,53,7

Question

The digits indicated by * in 3422213** so that this number is divisible by 99 areOptions1,94,65,53,7

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Solution

To find the digits that make the number 3422213** divisible by 99, we need to understand the divisibility rule for 99. A number is divisible by 99 if and only if the number is divisible by both 9 and 11.

  1. Divisibility rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Let's add the digits of the number 3422213**: 3 + 4 + 2 + 2 + 2 + 1 + 3 = 17. We don't know the last two digits, so let's call them a and b. The sum of the digits is now 17 + a + b. For this to be divisible by 9, a + b must be 2 (because 17 + 2 = 19, which is divisible by 9).

  2. Divisibility rule for 11: A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11. Let's calculate this for the number 3422213**: Sum of digits in odd positions = 3 + 2 + 2 + 3 + a = 10 + a. Sum of digits in even positions = 4 + 2 + 1 + b = 7 + b. The difference between these sums is (10 + a) - (7 + b) = 3 + a - b. For this to be divisible by 11, a - b must be 8 (because 3 + 8 = 11, which is divisible by 11).

From these two rules, we have two equations: a + b = 2 and a - b = 8. Solving these equations gives a = 5 and b = -3. However, b cannot be negative in this context, so there is no solution that makes the number 3422213** divisible by 99. Therefore, none of the given options are correct.

This problem has been solved

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