What is the largest four-digit number that is exactly divisible by 88 without leaving any remainder?

Introduction / Context:
This problem is about finding the greatest four-digit number that is exactly divisible by a given divisor, here 88. Such questions appear frequently in number system topics and are useful to test understanding of multiples and division. The key idea is to find the largest multiple of 88 that does not exceed the largest four-digit number, which is 9999.

Given Data / Assumptions:

• The largest four-digit number is 9999.
• We need a number of the form 88 * k that is less than or equal to 9999.
• The number must be exactly divisible by 88, meaning remainder 0.
• We must find the largest such multiple.

Concept / Approach:
To find the largest four-digit number divisible by 88, we divide 9999 by 88 and consider the integer part of the quotient. That quotient multiplied by 88 gives the largest multiple of 88 not exceeding 9999. We do not need the remainder, except to confirm that 9999 itself is not divisible by 88. This approach avoids checking each option individually and directly uses division to locate the required multiple.

Step-by-Step Solution:
Step 1: Take the largest four-digit number, which is 9999. Step 2: Divide 9999 by 88 to find how many full multiples of 88 fit into 9999. Step 3: Perform the division: 9999 / 88 gives a quotient of 113 and some remainder. Step 4: Compute 88 * 113 = 9944. Step 5: Next multiple would be 88 * 114 = 10032, which exceeds 9999 and is therefore not a four-digit number. Step 6: Hence, 9944 is the largest four-digit number divisible by 88. Step 7: Verify that 9944 is indeed divisible by 88 by dividing: 9944 / 88 = 113 exactly, with remainder 0.

Verification / Alternative check:
We can also look at the options. Option (a) 8888 divided by 88 equals 101, which is a valid multiple but smaller than 9944. Option (b) 9999 divided by 88 gives a non-integer quotient with a remainder, so it is not exactly divisible. Option (d) 9988 divided by 88 equals 113 with remainder ... wait, computing clearly shows that 88 * 113 is 9944, so 9988 is not exactly divisible by 88. Therefore, among the options, only 9944 is both a four-digit number and the largest that is a multiple of 88 below 9999.

Why Other Options Are Wrong:
Option (a) 8888: A correct multiple of 88 but not the largest possible four-digit multiple.
Option (b) 9999: Not divisible by 88, as the division leaves a remainder.
Option (d) 9988: Not equal to 88 times an integer, so it is not exactly divisible by 88.

Common Pitfalls:
A common mistake is to choose 9999 or the numerically largest option without checking divisibility. Some students also miscalculate 88 times the quotient or misread the remainder. Using accurate division and verifying one or two key multiples helps ensure the correct result.

Final Answer:
The largest four-digit number exactly divisible by 88 is 9944.