What is the Smallest Number Subtracted from 8112 to Make It Exactly Divisible by 99?

In this article, we will explore how to find the smallest number that should be subtracted from 8112 to make it exactly divisible by 99. This question involves a mix of basic number theory and modular arithmetic, which are fundamental concepts for understanding divisibility rules and remainders.

Step-by-Step Solution

To solve this problem, we can follow a systematic approach. The first step is to find the remainder of 8112 when divided by 99. Let's start by performing the division and finding the remainder:

1. First, divide 8112 by 99:

8112 ÷ 99 ≈ 82 (integer part of the division)Next, calculate the product of 82 and 99: 82 × 99 8118

2. Now, find the remainder by subtracting the product from 8112:

8112 - 8118 -6

3. Since a remainder cannot be negative, we convert it to a positive value using the modulo operation:

8112 mod 99 99 - 6 93

Therefore, the remainder when 8112 is divided by 99 is 93. To make 8112 exactly divisible by 99, we need to subtract this remainder from 8112.

The smallest number to subtract from 8112 to make it exactly divisible by 99 is 93.

Further Exploration

Let's consider if we can find even smaller numbers by repeatedly subtracting 99 from 93:

93 - 99 -6 (negative number, not useful for our case)Since 93 is the smallest positive remainder, any further subtraction by 99 will result in negative numbers, which do not provide a meaningful solution to the problem of exact divisibility.

Thus, the smallest positive integer that needs to be subtracted from 8112 to make it exactly divisible by 99 is 93.

Alternative Methods

Here are some alternative methods to find the solution:

Method 1: Direct Subtraction from 99

1. Calculate 8112 ÷ 99 81.939393 (RECURSIVE).

2. The fractional part is 0.93939393.3. Multiply the fractional part by 99: 0.93939393 × 99 93.

Thus, the answer is 93. This method works because the fractional part represents the remainder, and multiplying it by 99 gives us the smallest number to subtract.

Method 2: Identifying Nearest Multiple

1. Find the nearest multiple of 99 that is less than 8112.

2. 8100 is the nearest multiple of 99 (81 × 99).3. Subtract 8112 - 8100 112.4. Calculate 8112 - 112 8000.

5. 8000 ÷ 99 80 (exact division).

Hence, 93 is the smallest number to subtract.

Method 3: Using Fractional Approach

1. Divide 8112 by 99: 8112 ÷ 99 81.939393.

2. The fractional part is 0.939393.3. Multiply the fractional part by 99: 0.939393 × 99 93.

4. Subtract this value from 8112: 8112 - 93 8019.

5. 8019 ÷ 99 81 (exact division).

Hence, 93 is indeed the smallest number to subtract.

Conclusion

This problem demonstrates the importance of understanding the rules of modular arithmetic and divisibility. The smallest number that must be subtracted from 8112 to make it exactly divisible by 99 is 93. The methods provided offer different perspectives on solving this problem, and each method can be used to verify the solution.