Smallest 3-Digit Number With Remainder 3 When Divided By 8
Introduction to the Problem
In the realm of mathematics, we often encounter intriguing problems that challenge our understanding of numbers and their properties. One such problem involves finding the smallest three-digit number that leaves a specific remainder when divided by another number. In this case, we are tasked with identifying the smallest three-digit number that gives a remainder of 3 when divided by 8. This problem combines the concepts of division, remainders, and number ranges, providing an excellent exercise in mathematical reasoning. Understanding the intricacies of remainders and how they behave in division is crucial for solving this type of problem. The three-digit number constraint adds another layer of complexity, requiring us to consider the range of numbers we are working with. By carefully analyzing the conditions given, we can develop a systematic approach to finding the solution.
Understanding Remainders and Division
Before diving into the solution, it's essential to have a firm grasp of the concept of remainders in division. When one number (the dividend) is divided by another number (the divisor), the result is a quotient and a remainder. The remainder is the amount "left over" after the division is performed as many times as possible without resulting in a fraction. For example, when 27 is divided by 8, the quotient is 3, and the remainder is 3 because 8 goes into 27 three times (8 * 3 = 24), leaving 3 as the remainder (27 - 24 = 3). In mathematical terms, we can express this relationship as: Dividend = (Divisor * Quotient) + Remainder. This equation is fundamental to understanding division and remainders, and it will be instrumental in solving our problem. The remainder is always a non-negative integer and must be less than the divisor. If the remainder is 0, it means the dividend is perfectly divisible by the divisor. Understanding these basic principles of division and remainders is essential for tackling problems involving number theory and modular arithmetic. It allows us to break down complex problems into simpler, more manageable parts.
Identifying the Range of Three-Digit Numbers
To solve our problem effectively, we need to define the range of three-digit numbers we are working with. The smallest three-digit number is 100, and the largest is 999. Our target number must fall within this range. This constraint helps us narrow down the possibilities and provides a starting point for our search. We know that the number we seek must be greater than or equal to 100, and this information is crucial for establishing a lower bound. Without this constraint, we might end up with a two-digit or even a one-digit number, which would not satisfy the problem's conditions. The upper bound of 999 is less critical in this specific problem, but it's still important to keep in mind as it defines the overall scope of our search. By focusing on the range of three-digit numbers, we can apply targeted techniques to find the solution more efficiently. This range provides a clear framework for our calculations and ensures that we find a number that meets the specified criteria.
Finding the Smallest Multiple of 8 Close to 100
Our strategy involves finding the smallest multiple of 8 that is close to 100, the lower bound of our three-digit range. To do this, we can divide 100 by 8. The result is 12.5. Since we are looking for a whole number multiple, we take the integer part of this result, which is 12, and multiply it by 8: 12 * 8 = 96. This tells us that 96 is the largest multiple of 8 that is less than 100. This step is crucial because it gives us a baseline from which to work. We know that the number we are looking for must leave a remainder of 3 when divided by 8, so we need to adjust this multiple accordingly. By identifying the multiple of 8 closest to 100, we can efficiently find the desired number without having to test every three-digit number individually. This approach utilizes the properties of multiples and remainders to streamline the solution process. It demonstrates a practical application of number theory concepts in problem-solving.
Adding the Remainder to Find the Solution
Now that we have found the largest multiple of 8 less than 100 (which is 96), we need to incorporate the remainder requirement. The problem states that the number must leave a remainder of 3 when divided by 8. Therefore, we add the remainder (3) to the multiple we found: 96 + 3 = 99. However, 99 is not a three-digit number, so it doesn't meet the initial condition of our problem. This means we need to consider the next multiple of 8. To do this, we add 8 to our previous multiple (96), giving us 104. Then, we add the remainder: 104 + 3 = 107. This number is a three-digit number and leaves a remainder of 3 when divided by 8. To confirm that this is the smallest such number, we can check the previous potential solution (99), which we already determined was not a three-digit number. Therefore, 107 is indeed the smallest three-digit number that satisfies the given conditions. This step demonstrates the importance of carefully considering all conditions of the problem and iteratively refining our solution until all criteria are met.
Verifying the Solution
To ensure our solution is correct, we must verify that 107 indeed leaves a remainder of 3 when divided by 8. We perform the division: 107 ÷ 8 = 13 with a remainder of 3. This confirms that our solution satisfies the remainder condition. Additionally, we need to verify that 107 is the smallest three-digit number that meets this condition. We have already established that 99 (which is 96 + 3) is not a three-digit number. Therefore, 107 is the smallest three-digit number that leaves a remainder of 3 when divided by 8. This verification step is crucial in problem-solving. It ensures that we have not made any errors in our calculations or reasoning. By systematically checking our solution against all the conditions of the problem, we can be confident in our answer. This process reinforces the importance of accuracy and attention to detail in mathematical problem-solving.
Conclusion: The Smallest Three-Digit Number
In conclusion, after carefully analyzing the problem and applying a systematic approach, we have successfully identified the smallest three-digit number that leaves a remainder of 3 when divided by 8. The number is 107. This problem demonstrates the application of fundamental mathematical concepts such as division, remainders, and number ranges. By understanding these concepts and employing a logical problem-solving strategy, we can tackle similar challenges effectively. The process of finding the solution involved several key steps: understanding the concept of remainders, identifying the range of three-digit numbers, finding the smallest multiple of 8 close to 100, adding the remainder to find a potential solution, and verifying the solution to ensure its accuracy. This systematic approach is valuable not only for solving mathematical problems but also for approaching challenges in various other fields. The ability to break down a complex problem into smaller, more manageable steps and to apply logical reasoning is a crucial skill for success in many areas of life. This exercise reinforces the importance of mathematical thinking and its practical applications.
