What is the largest four digit number that is exactly divisible by 88?

Introduction / Context:
This question is a standard divisibility and number range problem. We are asked to find the largest four digit number that is exactly divisible by 88. Four digit numbers range from 1000 to 9999, and the task is to locate the greatest multiple of 88 within this interval. This tests understanding of division with remainders and the relationship between multiples and ranges.

Given Data / Assumptions:

- Four digit numbers are from 1000 to 9999 inclusive. - We want the largest integer in this range that is a multiple of 88. - The answer options include several candidates near the upper end of the four digit range.

Concept / Approach:
To find the largest multiple of 88 less than or equal to 9999, we perform integer division of 9999 by 88. The quotient tells us how many full groups of 88 fit into 9999, and multiplying this quotient by 88 gives the largest multiple of 88 that does not exceed 9999. We do not need to check all the options by direct divisibility tests if we compute this value once accurately.

Step-by-Step Solution:
Step 1: Consider the upper limit of four digit numbers, which is 9999. Step 2: Compute the integer quotient when 9999 is divided by 88. Step 3: Perform division: 9999 ÷ 88 = 113 with some remainder, because 88 * 113 = 9944. Step 4: Compute 88 * 113 to confirm: 88 * 113 = 88 * 100 + 88 * 13 = 8800 + 1144 = 9944. Step 5: Since 9944 is less than 9999 and is exactly divisible by 88, it is a valid candidate. Step 6: The next multiple would be 88 * 114, which equals 10032, a five digit number and therefore outside the allowed four digit range. Step 7: Thus, 9944 is the largest possible four digit multiple of 88.

Verification / Alternative check:
We can quickly check the options by dividing them by 88. For 9944, 9944 ÷ 88 = 113 exactly. For 9768, dividing by 88 gives 111 with remainder, and for 8888, 8888 ÷ 88 is not an integer either. Option 9988 is larger than 9944, but 9988 ÷ 88 does not give a whole number. Therefore, no option larger than 9944 is a valid multiple of 88 within the four digit range, confirming our result.

Why Other Options Are Wrong:
Options B, C and D are either not multiples of 88 or fall outside the required definition of the largest four digit multiple. Specifically, they leave non zero remainders when divided by 88 or exceed the maximum allowed number when extended beyond 9999.

Common Pitfalls:
Some test takers may be tempted to pick 9988 simply because it is close to 9999, without checking divisibility. Others may try to work upward from a smaller multiple instead of using integer division from the upper bound. Systematically using quotient and product avoids guesswork and ensures a correct answer.

Final Answer:
The largest four digit number exactly divisible by 88 is 9944, corresponding to option A.