Find the largest five digit number that is exactly divisible by 89.

Introduction / Context:
This question involves divisibility and working with the upper limit of a range, here the set of all five digit numbers. You must determine the largest five digit number that is divisible by 89. This kind of problem appears frequently in competitive exams to test comfort with division, remainders, and reasoning about multiples near a boundary like 99999.

Given Data / Assumptions:

• We are looking at five digit numbers, from 10000 to 99999.
• We want the largest number in this range that is exactly divisible by 89.
• Exactly divisible means the remainder on division by 89 is zero.
• We assume standard integer arithmetic.

Concept / Approach:
The key idea is to find the greatest multiple of 89 that does not exceed 99999. This can be done by dividing 99999 by 89 and taking the integer part of the quotient. That integer quotient times 89 will be the largest multiple of 89 not greater than 99999. We do not need to test numbers from the options by trial if we perform this calculation correctly.

Step-by-Step Solution:
Step 1: Consider the maximum five digit number 99999. Step 2: Divide 99999 by 89 to find how many full multiples of 89 fit inside 99999. Step 3: The integer part of 99999 / 89 is 1123. This means 1123 is the largest integer k such that 89 * k ≤ 99999. Step 4: Compute the product 89 * 1123. Step 5: Multiplying gives 89 * 1123 = 99947. Step 6: Therefore 99947 is a multiple of 89 and it is less than 99999. Step 7: Any larger multiple would be 89 * 1124, which is 99947 + 89 = 100036, which exceeds the five digit limit. Step 8: Thus 99947 is the largest five digit number exactly divisible by 89.

Verification / Alternative check:
To verify, we can perform a direct division of 99947 by 89. Dividing gives a quotient of 1123 and a remainder of zero, confirming exact divisibility. If we try nearby numbers such as 99940, 99938, or 99939, and divide by 89, we will not get integer quotients. For example, if 99940 were divisible by 89, then 99940 divided by 89 would have to be an integer, but it is not. This confirms that among the options, 99947 is the only valid answer.

Why Other Options Are Wrong:
Numbers like 99940, 99938, and 99939 are close to 99947 but they are not multiples of 89. They produce non zero remainders when divided by 89. The option 99858 is also a multiple of 89, but it is smaller than 99947, so it cannot be the largest five digit multiple. Therefore, only 99947 satisfies both conditions of being five digits and being the largest number exactly divisible by 89.

Common Pitfalls:
A common error is to guess based on the last digits and not check divisibility properly, or to try dividing each option instead of reasoning from the top of the range. Another pitfall is rounding the quotient incorrectly or miscalculating the product of 89 and the quotient. Carefully performing the division and checking the final product avoids these errors. Remember that the standard method is to take the floor of the division of the upper bound by the divisor, then multiply back to obtain the largest multiple within the allowed range.

Final Answer:
The largest five digit number exactly divisible by 89 is 99947.