Finding The Smallest 4-Digit Number With A Digit Sum Of 18

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Hey guys! Let's dive into a fun little math puzzle. The task at hand is to find the smallest 4-digit number that has a sum of its digits equal to 18. And, as a bonus, we'll write that number out in Roman numerals! Sounds like a blast, right? This isn't just about crunching numbers; it's a bit of logical thinking too. We're going to break down how to approach this, step by step, so you can follow along easily, whether you're a math whiz or just brushing up on your skills. We'll also touch on why this specific number is considered the smallest possible, and of course, the Roman numeral conversion. Ready? Let's get started.

To begin, let's clarify what we're dealing with. A 4-digit number is any number that has four places: thousands, hundreds, tens, and units. Think of it like this: _ _ _ _. Each blank represents a digit. The smallest 4-digit number possible is 1000. But the key is that the sum of all the digits in our target number must equal 18. For example, in the number 1000, the sum is just 1 (1+0+0+0 = 1). So, 1000 is definitely not the answer. We need to find a number that fits the bill – a number that is as small as possible, while still having its digits add up to 18.

So how do we do this? We need to consider how to make the number as small as we can while still satisfying the sum condition. The key is to keep the thousands place as small as possible. The smallest digit we can use in the thousands place is '1'. If we started with '2' in the thousands place, the number would already be larger than anything starting with '1'. So, our number starts with 1: 1 _ _ _. Now, we have to make sure that the remaining three digits add up to 17 because our goal is that the sum of all the digits is 18. If the first digit is 1, we need the other digits to sum up to 17 (18 - 1 = 17). Now, what's the smallest we can make the hundreds place? We ideally want to use a 0 to keep the number as small as possible. However, since 1 + 0 + 0 + 0 = 1, which is not 18. It would mean the other digits must sum to 17. So, we'll need to put some digits in those remaining spots. In order to make the number as small as possible, we want the next digit (hundreds place) to be as small as possible, but still allow us to make the rest of the number sum to 18. So we can put a '7' in the hundreds place (1 _ _ _ and since 1 + 7 = 8, we need 10 more (18 - 8 = 10). Finally, to minimize the tens and units place, let's put a '9' in the tens place and a '0' in the unit place; this is not the smallest number because the order of the digits matter! After all this, the number becomes 1790.

Step-by-Step Breakdown to Find the Answer

Alright, let's go through this systematically. We're not just guessing and checking here; we're using a clear strategy. This approach ensures we find the absolute smallest number that meets the conditions. It is all about precision and a methodical way of thinking.

  1. Start with the smallest thousands digit: The smallest digit for the thousands place is 1. This makes our number look like this: 1 _ _ _.
  2. Maximize the remaining digits: Our target sum is 18, and we already used 1. That leaves us with 17 more to make up. To keep the number as small as possible, let's put a 7 in the hundreds place (1 + 7 = 8). Since, 18 - 8 = 10, we have 10 more to make up. To minimize the number, we use the largest number possible in the tens place to make the number as small as possible (1+ 7 + 9 = 17). Finally, the units place becomes 0 (1 + 7 + 9 + 0 = 17). So, our number becomes 1790.
  3. Verify: Now, let's double-check. The digits are 1, 7, 9, and 0. Their sum is 1 + 7 + 9 + 0 = 17! Oh snap, it doesn't add up to 18. What did we do wrong? We want a digit sum of 18. So, let's switch the tens and units place: 1 + 7 + 0 + 9 = 17! This is still not correct. The tens place must be 9 to be small and to make the sum equal to 18. With that in mind, the order must be in ascending order. Then we should switch the hundreds and tens digit: 1 + 0 + 9 + 8 = 18! The smallest number we can get is 1980. The digits add up to 18 (1+9+8+0=18), so it meets our first requirement. The question is: is it the smallest number possible? The answer is yes. Now we can finally write it in Roman Numerals!

So, the smallest 4-digit number with a digit sum of 18 is 1980.

Writing the Answer in Roman Numerals

Here's where the fun really starts! Now that we've got our answer, 1980, let's transform it into Roman numerals. Just a quick refresher: Roman numerals use letters to represent numbers. Here's a little cheat sheet:

  • I = 1
  • V = 5
  • X = 10
  • L = 50
  • C = 100
  • D = 500
  • M = 1000

To write 1980 in Roman numerals, we'll break it down by place value:

  • 1000 = M
  • 900 = CM (Remember, 900 is 1000 - 100, so it's CM)
  • 80 = LXXX (80 = 50 + 10 + 10 + 10, so it's LXXX)

So, putting it all together, 1980 in Roman numerals is MCMLXXX.

Why Is 1980 the Smallest?

We've already touched on this, but let's make sure it's crystal clear. The reason 1980 is the smallest is because of how we constructed it. We started with the smallest possible digit in the thousands place (1). Then, to keep the number as small as possible, we wanted to have the next digit to be 0, but, the question is that the sum is 18. If we use 0, the rest of the digits must add up to 17! To keep the number small, we can use 9, then 8, and 0. In order to make sure that the numbers is as small as possible, the numbers must be in ascending order! Then, our answer is 1980. We can't use any other 4-digit number smaller than this one and still get the sum of the digits to be 18. Any other combination would either be smaller but not add up to 18, or add up to 18 but be larger. This is because of how the place values work. When you change the value in the thousands place, it impacts the overall size the most. So, by making sure that thousands place is at its lowest value while following the other conditions, we are guaranteed to find the smallest possible 4-digit number.

Conclusion

So, there you have it, guys! We found the smallest 4-digit number with a digit sum of 18 – which is 1980 – and then we wrote it out in Roman numerals: MCMLXXX. It was a great exercise in logical thinking and number manipulation. We broke down the problem into simple steps, making it easy to understand and fun to solve. We not only got the answer but also understood why it's the correct answer. Math can be like a fun puzzle, you just have to know how to look at it! Hope you enjoyed it, and happy number crunching!