What is the greatest four digit number that is exactly divisible by 81?

Introduction / Context:
This problem tests divisibility and the idea of finding the greatest multiple of a given number within a specified range. Here the range is all four digit numbers, from 1000 to 9999, and the divisor is 81.

Given Data / Assumptions:
- We are asked for the largest four digit integer that is divisible by 81.
- The largest four digit number overall is 9999.
- We need the greatest integer m such that m ≤ 9999 and 81 divides m exactly.

Concept / Approach:
To find the greatest multiple of a number d that does not exceed N, we divide N by d, take the integer part of the quotient, and multiply back by d. That is, if q = floor(N ÷ d), then the required multiple is q * d. Here d = 81 and N = 9999.

Step-by-Step Solution:
Let N = 9999 and d = 81. Compute the integer quotient q = floor(9999 ÷ 81). Perform the division: 81 × 123 = 9963 and 81 × 124 = 10044 which exceeds 9999. Therefore q = 123, since 124 gives a product above the four digit range. The greatest multiple is q * d = 123 × 81 = 9963.

Verification / Alternative Check:
Check that 9963 is four digit and divisible by 81. Dividing 9963 by 81 gives 123 exactly. Any larger four digit number than 9963 would either not be divisible by 81 or would be above 9999. For example, 9993 ÷ 81 does not yield an integer quotient, so 9993 is not acceptable as the required multiple.

Why Other Options Are Wrong:
9993 is not divisible by 81, since it does not give an integer quotient.
9936 and 9918 are smaller than 9963 and therefore cannot be the greatest multiple if a larger valid multiple exists.
9900 is also divisible by several numbers but is smaller than 9963 and thus cannot be the largest four digit multiple of 81.

Common Pitfalls:
Common errors include checking only the options without using the floor division idea, which can lead to missed or misjudged calculations. Another pitfall is assuming that the largest number ending with certain digits would be divisible, without verifying by exact division. Using the systematic method of floor division is more reliable and efficient.

Final Answer:
The largest four digit number exactly divisible by 81 is 9963.