Finding 3-Digit Even Numbers With Repeated 5s A Math Challenge

by Scholario Team 63 views

Hey everyone! Let's dive into a cool math problem today. We're going to figure out how to find all the 3-digit even numbers that have the digit 5 repeated exactly twice. This isn't just about crunching numbers; it's about thinking logically and systematically. So, grab your mental gears and let's get started!

Understanding the Problem: What Are We Looking For?

Before we jump into solving, let’s make sure we really understand what the question is asking. We need to find numbers that:

  • Are 3 digits long (meaning they are between 100 and 999).
  • Are even (so they end in 0, 2, 4, 6, or 8).
  • Have the digit 5 appear exactly two times.

This might sound a bit tricky, but by breaking it down, we can tackle it step by step. The key here is organized thinking. We don't want to miss any numbers, and we certainly don't want to count the same number twice!

To ensure we cover all possibilities, let's consider the different positions where the digit 5 can appear. We'll explore each case systematically to build our list of numbers. Think of it like a detective solving a case – we're piecing together clues (the conditions of the problem) to find our solution (the numbers themselves).

Remember, even numbers are our focus, so the last digit plays a crucial role. This constraint will help us narrow down our options and make the search more manageable. So, let's put on our math hats and get to work! We're not just finding numbers; we're sharpening our problem-solving skills along the way.

Case-by-Case Analysis: Where Can Those 5s Go?

Okay, so let's break this down into cases to make sure we're super organized. We've got three digits to play with, and the digit 5 needs to show up twice. This means we need to think about the possible spots those 5s can occupy. Think of it like this: we have three slots (hundreds, tens, and units), and we need to place two 5s in them.

Here are the three main scenarios we need to consider:

  1. 5 is in the hundreds and tens place: This means our number looks like 55_. The blank space is where the last digit goes, and it must be an even number (0, 2, 4, 6, or 8) to make the whole number even.
  2. 5 is in the hundreds and units place: Now our number looks like 5_5. The middle digit needs to be something other than 5, and to keep the number even, the 5 in the units place is already taken care of.
  3. 5 is in the tens and units place: This gives us _55. The first digit (hundreds place) can be anything from 1 to 9 (it can't be 0 because then it wouldn't be a 3-digit number), but remember, it also can't be 5.

By looking at these cases separately, we make sure we don't miss any possibilities. It's like sorting your socks – you group them together to make sure you have all the pairs! This methodical approach is what makes problem-solving easier and more accurate. In each case, we'll consider the restrictions (like the number needing to be even) to figure out the possible digits for the remaining spot. So, let's roll up our sleeves and dive into each case one by one.

Case 1: 5 in the Hundreds and Tens Place (55_)

Alright, let's tackle the first case: numbers that look like 55_. We know the hundreds and tens digits are both 5, so the only digit we need to figure out is the units digit. Now, remember the golden rule: our number needs to be even! This is super important.

What digits make a number even? You got it – 0, 2, 4, 6, and 8. So, we can just plug each of these into the blank spot and see what we get:

  • 550
  • 552
  • 554
  • 556
  • 558

That's it! We've found all the even numbers that fit this pattern. There are five numbers in total. This case was pretty straightforward, right? The even number rule really helped us narrow down the possibilities.

It's almost like we're building a puzzle, and each case is a different piece. We've successfully placed this piece, and now we can move on to the next one. This methodical approach not only helps us find the answer but also makes the process less overwhelming. We're not trying to solve everything at once; we're taking it one step at a time. So, with this case wrapped up, let's move on to the next scenario and see what numbers we can find!

Case 2: 5 in the Hundreds and Units Place (5_5)

Okay, time for case number two! This time, we're looking at numbers in the form 5_5. We know the hundreds digit is 5 and the units digit is also 5. That means the tens digit is the one we need to figure out. But there’s a catch! Remember, the digit 5 can only appear twice in our number. So, whatever digit we put in the blank space, it can't be a 5.

Also, recall that we're looking for even numbers. But wait a minute… Our number already ends in 5! And numbers ending in 5 are always odd. This is a crucial observation. It tells us something really important: there are no even numbers in this case!

Think about it – if a number ends in 5, it's automatically odd. No matter what digit we put in the tens place, it won't change the fact that the number is odd. This might seem like a dead end, but it's actually a valuable piece of information. We've explored this possibility, and we know it doesn't give us any solutions.

This is a perfect example of how problem-solving isn't always about finding answers right away. Sometimes, it's about ruling out possibilities. By recognizing that this case can't produce any even numbers, we save ourselves time and effort. We can confidently move on to the next case knowing that we've thoroughly investigated this one. So, let's keep going! We're making progress, one case at a time.

Case 3: 5 in the Tens and Units Place (_55)

Alright, let's dive into our final case: numbers that have the form _55. This means the tens and units digits are both 5, and we need to figure out what goes in the hundreds place. Now, there are a couple of things we need to keep in mind here.

First, the hundreds digit can't be 0. If it were, we wouldn't have a 3-digit number – it would just be a 2-digit number. Also, remember our rule: the digit 5 can only appear twice. So, the hundreds digit also can't be 5.

But here’s the kicker, guys... Just like in the last case, we're looking for even numbers. And any number ending in 5 is odd! This means that no matter what digit we put in the hundreds place (as long as it's not 0 or 5), the number will always be odd.

So, what does this tell us? It's the same situation as before: there are no even numbers that fit this pattern. We've hit another dead end, but that's okay! Knowing where not to look is just as important as knowing where to look.

This case highlights the importance of paying close attention to all the conditions of the problem. We can't just focus on the placement of the 5s; we also need to remember the even number requirement. By considering all the rules, we can quickly identify when a case won't lead to a solution. It's like a detective eliminating suspects – we're narrowing down the possibilities until we find the answer. So, with this case ruled out, let's take a look at the big picture and see what we've discovered.

Conclusion: Putting It All Together

Okay, guys, we've done it! We've explored all the possibilities and figured out the answer to our math challenge. We systematically broke down the problem into cases, analyzed each one, and discovered some important things along the way. Now, let's gather our findings and see what we've got.

Remember, we were looking for 3-digit even numbers that have the digit 5 repeated exactly twice. We considered three main cases:

  • Case 1: 5 in the hundreds and tens place (55_) – This gave us the numbers 550, 552, 554, 556, and 558.
  • Case 2: 5 in the hundreds and units place (5_5) – We found that there were no even numbers in this case because any number ending in 5 is odd.
  • Case 3: 5 in the tens and units place (_55) – Similarly, we discovered that there were no even numbers in this case either, for the same reason.

So, what's the final answer? The only even numbers that fit our criteria are the ones from Case 1: 550, 552, 554, 556, and 558. That's a total of five numbers!

This problem wasn't just about finding the right numbers; it was about developing a strategy for problem-solving. We learned the importance of breaking down complex problems into smaller, manageable cases. We also saw how crucial it is to pay attention to all the conditions of the problem (like the even number requirement) and how ruling out possibilities can be just as helpful as finding solutions. Great job, everyone! You tackled this math challenge like true pros. Keep practicing, and you'll become even better problem-solvers in the future!