Largest & Smallest 3-Digit Numbers With Remainder 8 (Divisible By 11)

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Hey guys! Today, we're diving into a fun math problem that involves finding the largest and smallest 3-digit natural numbers that give us a remainder of 8 when divided by 11. Sounds interesting, right? Let's break it down step by step and figure out how to solve it. We'll explore the concepts of remainders, divisibility, and how to identify numbers within a specific range that meet our criteria. So, buckle up and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand the question clearly. We're looking for two numbers:

  • The largest 3-digit number that, when divided by 11, leaves a remainder of 8.
  • The smallest 3-digit number that, when divided by 11, leaves a remainder of 8.

To tackle this, we need to understand what remainders are and how they work in division. Remember, the remainder is the amount left over after dividing one number by another. For example, when 20 is divided by 6, the quotient is 3 and the remainder is 2 because 6 goes into 20 three times (3 x 6 = 18), and there are 2 left over (20 - 18 = 2).

Also, keep in mind what 3-digit numbers are. They range from 100 (the smallest) to 999 (the largest). We need to find numbers within this range that fit our remainder condition. This involves understanding divisibility rules and how to manipulate numbers to get the desired remainder. We'll use a combination of division and a bit of strategic thinking to pinpoint our answers.

Finding the Largest 3-Digit Number

Okay, let's start with finding the largest 3-digit number that fits our condition. Here’s how we can do it:

  1. Identify the largest 3-digit number: This is 999.
  2. Divide 999 by 11: 999 ÷ 11 = 90 with a remainder of 9.
  3. Analyze the remainder: We got a remainder of 9, but we want a remainder of 8. This means 999 is too big. We need to go down a bit.
  4. Subtract the difference: To get a remainder of 8, we need to subtract 1 from 999 (since 9 - 8 = 1). So, 999 - 1 = 998.
  5. Check our answer: Let's divide 998 by 11. 998 ÷ 11 = 90 with a remainder of 8. Bingo! That's exactly what we wanted.

So, the largest 3-digit number that leaves a remainder of 8 when divided by 11 is 998. See how we started with the biggest possible number and worked our way down until we found the one that matched our remainder criteria? This approach helps us efficiently narrow down the possibilities.

Finding the Smallest 3-Digit Number

Now, let's find the smallest 3-digit number that gives us a remainder of 8 when divided by 11. We'll use a similar approach, but this time we'll start from the smallest 3-digit number and work our way up.

  1. Identify the smallest 3-digit number: This is 100.
  2. Divide 100 by 11: 100 ÷ 11 = 9 with a remainder of 1.
  3. Analyze the remainder: We got a remainder of 1, but we need a remainder of 8. This means 100 is too small. We need to find a number that, when divided by 11, leaves a remainder of 8.
  4. Calculate the difference: To get a remainder of 8, we need to figure out how much to add to 100. The difference between the desired remainder (8) and the current remainder (1) is 7. However, directly adding 7 to 100 won't give us the correct number because it doesn't account for the multiples of 11.
  5. Find the correct number: Instead, we need to think about what number, when divided by 11, would be closest to 100. We know that 11 x 9 = 99, which is very close to 100. To get a remainder of 8, we need to add 8 to the multiple of 11. So, 99 + 8 = 107.
  6. Check our answer: Let's divide 107 by 11. 107 ÷ 11 = 9 with a remainder of 8. Perfect!

Therefore, the smallest 3-digit number that leaves a remainder of 8 when divided by 11 is 107. We started with the smallest possibility and adjusted it based on the desired remainder and the divisor. This method ensures we find the absolute smallest number that meets the conditions.

Putting It All Together

So, we've successfully found both the largest and smallest 3-digit numbers that leave a remainder of 8 when divided by 11:

  • Largest number: 998
  • Smallest number: 107

This problem highlights the importance of understanding remainders and how they behave in division. By combining basic division with a bit of logical thinking, we can solve these types of problems efficiently. It’s all about breaking down the problem into smaller, manageable steps and applying the right concepts.

Tips for Solving Similar Problems

If you encounter similar problems in the future, here are a few tips that might help:

  • Understand the Basics: Make sure you have a solid understanding of division, remainders, and divisibility rules. These are the building blocks for solving these problems.
  • Start with Extremes: Begin with the largest or smallest possible number in the range and work your way towards the solution. This helps narrow down the possibilities.
  • Check Your Answers: Always verify your answers by performing the division and checking the remainder. This ensures accuracy.
  • Look for Patterns: Sometimes, there might be patterns in the numbers that can help you solve the problem more quickly. For example, you might notice that numbers with the same remainder are a fixed distance apart.
  • Practice Makes Perfect: The more you practice, the better you'll become at solving these types of problems. Try different variations and challenge yourself.

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