Smallest Distinct Digit Sum: A Math Puzzle!

Hey everyone! Let's dive into a fun math problem that involves finding the sum of the smallest numbers with distinct digits. We'll be looking at both 3-digit and 2-digit numbers, so get your thinking caps on! This isn't just about crunching numbers; it’s about understanding place value and how digits work together. So, let’s break it down and make sure we all get it. Whether you’re a math whiz or just brushing up your skills, this is going to be a piece of cake – a delicious math cake, that is!

Understanding the Basics of Distinct Digits

Before we jump into solving the problem, let’s quickly clarify what we mean by “distinct digits.” When we say a number has distinct digits, it simply means that no digit is repeated within that number. For example, the number 123 has distinct digits because 1, 2, and 3 are all different. However, the number 122 does not have distinct digits because the digit 2 is repeated. Understanding this concept is super crucial for solving our main problem, as it forms the foundation for identifying the smallest numbers that fit our criteria. So, keep this definition in mind as we move forward – it’s the key to unlocking the solution!

What Are Distinct Digits?

So, what exactly are distinct digits? Well, think of it like this: each digit in the number has to be unique, like snowflakes! No two digits can be the same. For instance, in the number 456, each digit (4, 5, and 6) is different, making them distinct. But if we had a number like 454, the digit 4 is repeated, so they're not distinct in that number. This idea is super important when we're trying to find the smallest numbers with this rule. We need to make sure we're not accidentally using the same digit twice, or else our number won't fit the bill! Got it? Great, let's keep going!

Why Distinct Digits Matter in This Problem

Now, you might be wondering, “Why are we even bothering with distinct digits?” Well, in this particular problem, it's the secret ingredient! The whole challenge revolves around finding the smallest numbers under this condition. If we didn't have the distinct digits rule, finding the smallest 3-digit or 2-digit number would be a piece of cake (spoiler alert: it would just be 100 and 10). But, by adding this little twist, we're making things a bit more interesting and really testing our understanding of number values. It pushes us to think critically about how digits work together to form numbers, and how the position of each digit affects the overall value. So, distinct digits aren't just a random condition; they're what make this problem a fun brain-teaser!

Finding the Smallest 3-Digit Number with Distinct Digits

Okay, let's get down to business and find that smallest 3-digit number with distinct digits! This is where we put our thinking caps on and really consider how numbers are structured. Remember, we want the tiniest possible number, but each digit has to be different. So, where do we even start? Well, let’s think about the hundreds place first. What’s the smallest digit we can put there? And how does that choice affect the rest of the number? Let’s break it down step by step to make sure we nail this!

Starting with the Hundreds Place

Let’s kick things off with the hundreds place – that's the first digit in our 3-digit number. To make the number as small as possible, we want to use the smallest digit we can. Now, you might be thinking, “Zero!” But hold on a second… if we put a zero in the hundreds place, it wouldn't be a 3-digit number anymore, would it? It would just be a 2-digit (or even 1-digit) number. So, zero is out! What's the next smallest digit? That’s right, it’s 1! So, we're starting our number with a 1 in the hundreds place. This is a great start! We’ve got the first piece of our puzzle in place. Now, let’s move on to the next digit and see how we can keep the number as small as possible while still keeping those digits distinct.

Moving to the Tens Place

Alright, we've got a 1 in the hundreds place, so let's tackle the tens place. We want to keep our number as small as possible, and we can't repeat any digits. So, what's the smallest digit we can use here? Remember, we already used 1, so we can't use that again. But, we can use 0! Zero is smaller than any other digit, and since it's not the first digit, it's totally fair game. So, we've got 1 in the hundreds place and 0 in the tens place – our number is shaping up nicely! We're one step closer to finding the smallest 3-digit number with distinct digits. Now, let's finish strong with the ones place!

Finishing with the Ones Place

Okay, we're on the home stretch! We've got 1 in the hundreds place and 0 in the tens place, so what goes in the ones place? We need a digit that's different from both 1 and 0. So, what's the smallest digit we haven't used yet? You guessed it – it's 2! So, we pop a 2 in the ones place, and voila! We've got our number. The smallest 3-digit number with distinct digits is 102! See? That wasn't so bad, was it? By breaking it down place by place and thinking carefully about our options, we were able to find the answer without any stress. Now, let's move on to the next part of our problem and find the smallest 2-digit number with distinct digits.

Finding the Smallest 2-Digit Number with Distinct Digits

Now that we've conquered the 3-digit number, let's set our sights on the smallest 2-digit number with distinct digits. This one is a bit simpler, but we still need to think it through! We're looking for the smallest possible number that has two different digits. So, how do we approach this? Just like before, let's start by thinking about the tens place and then move on to the ones place. What’s the smallest digit we can put in the tens place to make our number nice and small? Let’s figure it out!

Starting with the Tens Place (Again!)

Just like before, let's start with the tens place. This is the first digit in our 2-digit number, and we want it to be as small as possible. Can we use 0? Nope! If we put a 0 in the tens place, it wouldn't be a 2-digit number anymore; it would just be a single-digit number. So, zero is out of the question for the tens place. What's the next smallest digit? You got it – it's 1! So, we're putting a 1 in the tens place. We're off to a great start! Half of our number is already figured out. Now, let's finish strong and find the right digit for the ones place.

Finishing with the Ones Place

Alright, we've got a 1 in the tens place, so let's move on to the ones place. We need a digit that's different from 1 to keep our digits distinct. What's the smallest digit we haven't used yet? It's 0! Zero is smaller than all the other digits, and we haven't used it yet, so it's the perfect fit for our ones place. We pop a 0 in there, and boom! We've got it. The smallest 2-digit number with distinct digits is 10! See how easy that was? By starting with the smallest possible digit for each place, we quickly found our answer. Now, for the grand finale – let's add these two numbers together and solve the problem!

Adding the Numbers Together

Okay, we've done the hard work of finding the smallest 3-digit number with distinct digits (which is 102) and the smallest 2-digit number with distinct digits (which is 10). Now comes the fun part – adding them together! This is where we put our addition skills to the test. Remember, addition is all about combining the values of the numbers we're working with. So, we’re going to take 102 and 10 and see what they make together. Ready to do some math?

The Addition Process

So, let's add 102 and 10 together. We can do this just like we learned in school, by lining up the numbers and adding each column, starting from the right. We've got:

  102
+  10
-----

First, we add the digits in the ones place: 2 + 0 = 2. So, we write down a 2 in the ones place of our answer.

Next, we move to the tens place: 0 + 1 = 1. So, we write down a 1 in the tens place.

Finally, we look at the hundreds place. We only have a 1 in the hundreds place of 102, and nothing in the hundreds place of 10, so we just bring down the 1.

  102
+  10
-----
  112

And there we have it! 102 + 10 = 112. We've successfully added our two numbers together. High fives all around!

The Final Answer

Drumroll, please! After all that number sleuthing and adding, we've reached the final answer. The sum of the smallest 3-digit number with distinct digits (102) and the smallest 2-digit number with distinct digits (10) is… 112! 🎉 We did it! We took a tricky problem, broke it down into smaller steps, and solved it together. That's the power of math! So, next time you see a problem that looks intimidating, remember this: break it down, take it step by step, and you'll be amazed at what you can accomplish. You guys are awesome!