Smallest And Largest Four-Digit Natural Numbers With Two Identical Digits

by Scholario Team 74 views

Hey guys! Today, we're diving into a fun math problem that asks us to find the smallest and largest four-digit natural numbers that have exactly two identical digits. Sounds like a brain-teaser, right? Don't worry, we'll break it down step by step and make it super easy to understand. So, let's get started and explore the fascinating world of numbers!

Understanding the Basics

Before we jump into solving the problem, let's quickly recap some basic concepts. Remember, natural numbers are positive whole numbers starting from 1 (1, 2, 3, and so on). A four-digit number is any number between 1000 and 9999. And when we say "two identical digits," we mean that only two digits in the number are the same, while the other two are different.

Now, with these basics in mind, let's think about how to approach this problem. To find the smallest number, we'll want to use the smallest digits possible in the most significant places (the thousands and hundreds places). Conversely, to find the largest number, we'll want to use the largest digits possible in those places. This is a crucial strategy that will guide us in our quest for the solution. Keep this in mind as we move forward!

Finding the Smallest Four-Digit Number

Okay, let's tackle the first part of our challenge: finding the smallest four-digit natural number with exactly two identical digits. To minimize the number, we need to think strategically about which digits to use and where to place them. Remember, the smaller the digits in the higher places (thousands, hundreds), the smaller the overall number will be.

  1. The Thousands Digit: The smallest possible digit for the thousands place is 1 (we can't use 0 because that would make it a three-digit number). So, our number starts with 1.
  2. The Hundreds Digit: Now, let's think about the hundreds digit. To keep the number as small as possible, we should use 0. So, our number now looks like 10XX.
  3. The Identical Digits: We need two identical digits, and we've already used 1 and 0. To keep the number small, let's make the identical digits the smallest available, which is 0. So, we have 100X.
  4. The Last Digit: Finally, we need a digit for the units place that is different from 1 and 0. The smallest digit we can use is 2. This makes our number 1002.

Therefore, the smallest four-digit number with exactly two identical digits is 1002. See how we systematically worked through each digit to arrive at the smallest possible number? That's the key to solving problems like this!

Finding the Largest Four-Digit Number

Now that we've conquered the smallest number, let's switch gears and find the largest four-digit number with exactly two identical digits. The approach here is the opposite of what we did before: we want to use the largest digits possible in the most significant places.

  1. The Thousands Digit: To maximize the number, we start with the largest possible digit, which is 9. Our number begins with 9.
  2. The Hundreds Digit: We continue to maximize by using 9 again for the hundreds place. Our number now looks like 99XX.
  3. The Identical Digits: We already have two 9s, which fulfill the condition of having two identical digits. Now, we need to make the remaining digits as large as possible while ensuring they are different from 9.
  4. The Tens Digit: The largest digit we can use for the tens place, that isn't 9, is 8. So, our number is 998X.
  5. The Units Digit: Finally, for the units place, we need a digit that's different from 9 and 8. The largest such digit is 7. This gives us the number 9987.

So, the largest four-digit number with exactly two identical digits is 9987. Notice how we aimed for the highest possible digits in each position, leading us to the largest number that fits our criteria. Great job, guys!

Why This Approach Works

You might be wondering, why did we approach the problem this way? Why did we focus on the most significant digits first? Well, it all comes down to the place value system. Each digit in a number has a value that depends on its position. For example, in the number 1234, the digit 1 represents 1000, the digit 2 represents 200, the digit 3 represents 30, and the digit 4 represents 4.

The digits in the higher places (thousands, hundreds) have a much greater impact on the overall value of the number than the digits in the lower places (tens, units). Therefore, to minimize the number, we need to minimize the digits in the higher places. Conversely, to maximize the number, we need to maximize the digits in the higher places. This fundamental principle is the key to solving many number-related problems, so keep it in mind!

By strategically choosing digits for each place value, we ensure that we're constructing the smallest or largest possible number that meets the given conditions. This method provides a clear and logical pathway to the solution, making even seemingly complex problems manageable. It's like building with blocks – you start with the foundation and work your way up!

Let's Summarize

Alright, let's recap what we've learned today. We successfully found the smallest and largest four-digit natural numbers with exactly two identical digits. The smallest number is 1002, and the largest number is 9987. We achieved this by systematically considering each digit place and choosing the smallest or largest possible digit while adhering to the given conditions. We also discussed the importance of the place value system in determining the magnitude of a number.

Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. Think about what each digit represents and how it contributes to the overall value of the number. By applying this logical approach, you can conquer any number puzzle that comes your way! You've got this, guys!

Practice Makes Perfect

Now that you've seen how to solve this problem, it's time to put your skills to the test! Here are a few similar problems you can try:

  1. Find the smallest and largest three-digit numbers with exactly two identical digits.
  2. Find the smallest and largest four-digit numbers with exactly three identical digits.
  3. Find the smallest and largest five-digit numbers with exactly two identical digits.

Working through these problems will help you solidify your understanding of the concepts we've covered and develop your problem-solving abilities. Don't be afraid to experiment and try different approaches. The more you practice, the more confident you'll become in tackling numerical challenges. Remember, math is like a muscle – the more you exercise it, the stronger it gets!

Conclusion

So, guys, we've reached the end of our numerical adventure for today! We've explored the fascinating world of four-digit numbers and learned how to find the smallest and largest ones with specific characteristics. Remember, the key to success in these types of problems is a systematic approach, a solid understanding of place value, and a dash of logical thinking.

I hope you enjoyed this exploration and found it both informative and engaging. Keep practicing, keep exploring, and most importantly, keep having fun with numbers! Math is all around us, and the more we understand it, the better equipped we are to navigate the world. Until next time, happy problem-solving!