What is the greatest four-digit number that is exactly divisible by 93 without leaving any remainder?

Introduction / Context:
This problem is a classic example of divisibility and working with the largest or smallest numbers satisfying a condition. Here, we look for the largest four-digit number that is divisible by 93. Such questions are common in number system and arithmetic sections and help develop fluency in handling division, remainders, and bounds like four-digit limits.

Given Data / Assumptions:

• We are interested in four-digit numbers, that is, numbers from 1000 to 9999.
• The divisor is 93.
• We need the largest four-digit number that is exactly divisible by 93, which means the remainder must be zero.
• All calculations are done with integer arithmetic.

Concept / Approach:
To find the largest four-digit multiple of 93, we start from the largest four-digit number, 9999, and see how it behaves when divided by 93. If 9999 is not divisible by 93, the remainder tells us how much we must subtract to reach the nearest lower multiple. This uses the simple relation: number - remainder = nearest smaller multiple of the divisor. This is an efficient method compared to listing multiples one by one.

Step-by-Step Solution:
Step 1: The largest four-digit number is 9999.Step 2: Divide 9999 by 93 to find the quotient and remainder.Step 3: When 9999 is divided by 93, the quotient is 107 and the remainder is 48. So 9999 = 93 * 107 + 48.Step 4: To get the largest multiple of 93 that is not greater than 9999, subtract the remainder from 9999.Step 5: Compute 9999 - 48 = 9951.Step 6: Therefore, 9951 is a multiple of 93 and is less than or equal to 9999.Step 7: To confirm, divide 9951 by 93: 9951 / 93 = 107 exactly, so there is no remainder.

Verification / Alternative check:
An alternative viewpoint is to directly compute 93 * 108, which gives 10044. This is a five-digit number and therefore larger than any allowed four-digit number. So the previous multiple, 93 * 107, must be the largest four-digit multiple. As already computed, 93 * 107 = 9951, which matches our earlier result and confirms that 9951 is correct.

Why Other Options Are Wrong:
Option 9981: When divided by 93, it does not give an integer quotient; the remainder is not zero.
Option 9971: Also fails divisibility by 93 and produces a non-zero remainder.
Option 9961: Not a multiple of 93 and thus cannot be the correct answer.
Option 9924: Although closer, it is smaller than 9951 and still not the largest possible multiple if 9951 is a valid multiple.

Common Pitfalls:
One common mistake is to divide 1000 or 9999 roughly and then round incorrectly, leading to an off-by-one error in the multiple. Another pitfall is attempting to list many multiples manually, which is time-consuming and prone to arithmetic mistakes. A structured approach using division with remainder is faster and more reliable for exam conditions.

Final Answer:
The largest four-digit number divisible exactly by 93 is 9951.