Largest & Smallest 3-Digit Numbers With Remainder 10

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Hey guys! Let's dive into a cool math problem today. We're going to figure out the largest and smallest three-digit natural numbers that give us a remainder of 10 when divided by 17. Sounds like fun, right? This isn't just about crunching numbers; it's about understanding how numbers work and applying that knowledge. So, grab your thinking caps, and let's get started!

Understanding the Problem

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

  • The largest 3-digit number that, when divided by 17, leaves a remainder of 10.
  • The smallest 3-digit number that, when divided by 17, leaves a remainder of 10.

Think of it like this: if we divide our mystery number by 17, we'll get a whole number (the quotient) and a leftover of 10 (the remainder). To tackle this, we need to play around with multiples of 17 and see how they relate to the 3-digit range.

Why This Matters

Okay, so why bother with this kind of problem? Well, it's not just about getting the right answer. It's about:

  • Number Sense: Developing a feel for how numbers behave and relate to each other.
  • Problem-Solving Skills: Breaking down a problem into smaller, manageable steps.
  • Mathematical Reasoning: Using logic and deduction to arrive at a solution.

These skills are super useful, not just in math class, but in everyday life. So, let's get to it!

Finding the Smallest 3-Digit Number

Let's start by finding the smallest 3-digit number that fits our criteria. We need a number that's 10 more than a multiple of 17. The smallest 3-digit number is 100, so let's see how close we can get to 100 using multiples of 17.

Step-by-Step Approach

  1. Divide 100 by 17: 100 ÷ 17 = 5 with a remainder of 15.
  2. Interpret the Result: This tells us that 17 multiplied by 5 is less than 100. Specifically, 17 * 5 = 85.
  3. Find the Next Multiple: We need a multiple of 17 that's closer to 100. So, let's try 17 * 6 = 102.
  4. Subtract the Remainder Target: Now, remember we want a remainder of 10. So, we subtract 10 from our multiple: 102 - 10 = 92. This won't work because it’s less than 100.
  5. Find the Correct Multiple: The previous method led us astray; instead, we should find a multiple of 17 and add 10 to it. We already know 17 * 5 = 85. Let’s work with 17 * 6 = 102. Since 102 is the smallest multiple of 17 that's a 3-digit number, we subtract 17 to find the previous multiple: 102 - 17 = 85. So, we add 17 one more time: 17 * 6 = 102.
  6. Add the Remainder: Finally, add the desired remainder of 10: 102 + 10 = 112. Wait! This method is incorrect. We need to find a number that, when divided by 17, gives a remainder of 10, not a multiple plus 10.
  7. Correct Approach: We know that 17 * 5 = 85. If we add 10, we get 95, which is not a 3-digit number. Let's try 17 * 6 = 102. 102 - 17 = 85. So, we need a number greater than 85 that, when we add the remainder, it’s a multiple of 17. Let’s start trying the multiples of 17 starting from 6: 17 * 6 = 102. Now, we need to find a number where, when divided by 17, leaves a remainder of 10. So, if we express the number as 17x + 10, we need the smallest ‘x’ such that 17x + 10 >= 100.
  8. Solve the Inequality: 17x + 10 >= 100. Subtract 10 from both sides: 17x >= 90. Divide by 17: x >= 90/17 which is approximately 5.29. Since x has to be a whole number, the smallest possible value for x is 6.
  9. Calculate the Smallest Number: Substitute x = 6 into 17x + 10: (17 * 6) + 10 = 102 + 10 = 112.

So, the smallest 3-digit number that leaves a remainder of 10 when divided by 17 is 112. This method of finding the multiple and adding the remainder is key, guys! Make sure you understand why each step is important.

Finding the Largest 3-Digit Number

Now, let's tackle the other end of the spectrum and find the largest 3-digit number that leaves a remainder of 10 when divided by 17. We'll use a similar approach, but this time we'll start from the largest 3-digit number, which is 999.

Step-by-Step Approach

  1. Divide 999 by 17: 999 ÷ 17 = 58 with a remainder of 13.
  2. Interpret the Result: This means 17 * 58 is close to 999, but leaves a remainder of 13. We want a remainder of 10, so we need to adjust.
  3. Find the Appropriate Multiple: Since we got a remainder of 13, we are 3 more than our target remainder of 10. So we need to subtract 3 from 999 to get 996, which would be a multiple of 17 plus our desired remainder of 10. Let's verify this. Since 999 gives a remainder of 13, the number can be represented as 17 * 58 + 13. We want the remainder to be 10, so we need 17 * 58 + 10 + 3. If we subtract 3 from the original number 999, we have 996. Then the target number should be 996.
  4. Adjust to Get Remainder 10: Now let’s verify if the number 996 holds. If we subtract 10 from 999 to target a remainder of 10, the quotient would be 58 and the remainder would be 13. Then, what multiple of 17 would work? We know that 17 * 58 = 986. To get a remainder of 10, we want a number in the form of 17x + 10. So 17x + 10 <= 999, 17x <= 989, and x <= 989/17 = 58.176. So, the largest x will be 58. Let’s compute 17 * 58 + 10.
  5. Calculate the Largest Number: Substitute x = 58 into 17x + 10: (17 * 58) + 10 = 986 + 10 = 996.

Therefore, the largest 3-digit number that gives a remainder of 10 when divided by 17 is 996. Remember, working backwards from the largest number often makes finding the solution easier!

The Answers

Alright, guys, we've done the hard work! Let's recap our findings:

  • The smallest 3-digit number that leaves a remainder of 10 when divided by 17 is 112.
  • The largest 3-digit number that leaves a remainder of 10 when divided by 17 is 996.

Key Takeaways

This problem wasn't just about getting the right numbers; it was about the process. Here's what we learned:

  • Remainders are Key: Understanding what a remainder means is crucial for solving problems like this.
  • Multiples are Your Friends: Working with multiples of the divisor helps you narrow down the possibilities.
  • Start at the Extremes: Sometimes, starting with the smallest or largest possible number makes the problem easier to solve.
  • Check Your Work: Always double-check your answers to make sure they fit the given conditions.

Practice Makes Perfect

If you found this interesting, try similar problems with different divisors and remainders. You could even change the number of digits to make it more challenging. The more you practice, the better you'll get at these types of problems.

So, there you have it! We've successfully found the largest and smallest 3-digit numbers that leave a remainder of 10 when divided by 17. Keep practicing, and you'll become a math whiz in no time! Remember, guys, math is all about understanding the concepts and applying them in different ways. Keep exploring, keep learning, and most importantly, keep having fun!