Find The Digit That Makes 3 80_ Divisible By 8
Hey there, math adventurers! Ever stared at a number, a grand old number with a little secret waiting to be uncovered, and felt that tiny spark of curiosity? Well, get ready to have that spark ignite because we're about to embark on a super-duper, no-sweat mission to find a missing digit. We're talking about a number that's got a whole chunk of it spelled out: 380_. And guess what? This number, this magnificent 380-something, is dying to be perfectly divisible by 8. Yes, you heard me! Like a perfectly baked cake that just asks to be sliced and shared, this number wants to be divided by 8 without leaving a single crumb of a remainder. How exciting is that?
Now, you might be thinking, "Divisible by 8? That sounds complicated, like trying to herd cats in a vacuum cleaner!" But fear not, my friends! We're not going to pull out any complicated textbooks or wrestle with brain-bending algorithms. This is more like a fun scavenger hunt, a delightful puzzle where the prize is a perfectly happy, divisible-by-8 number. Think of it like trying to find the last piece of a jigsaw puzzle that just clicks into place. It's that satisfying!
Our mission, should we choose to accept it (and we totally should because it's awesome!), is to find the magical digit that goes into that little blank space: 380_. We've got our trusty number, 380, and we're missing just one teeny-tiny piece of the puzzle. It's like the number is wearing a disguise, and we need to reveal its true, divisible-by-8 form. And the best part? There's only one true hero for this job, one special digit that will make everything align perfectly. It’s like finding the perfect hat to go with your outfit – it just works!
So, how do we go about this daring quest? Well, we could get all scientific and start testing numbers one by one, right? We could try 3800, then 3801, then 3802, and so on. Imagine if you had to do that for a really, really big number! That would be like trying to count every single grain of sand on a beach. But here's the super-secret, fun-fact trick that makes our mission a breeze. We don't need to try all the numbers. We just need to focus on the end of the number. It's like looking at the last few steps of a dance routine to get the whole thing right.
The magic of divisibility by 8 lies in a little secret about the last three digits of a number. Yep, just the last three! So, for our number 380_, the part that truly matters for our 8-times-table adventure is the 80_. We need to find a digit that, when placed there, makes 80_ a number that 8 can gobble up without a fuss. Think of it this way: if the last three digits can dance perfectly to the tune of 8, the whole big number will be singing along too!
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Let’s play a little game. We know the first two digits of our crucial three-digit chunk are 8 and 0. So, we're looking at numbers like 800, 801, 802, and so on, all the way up to 809. We need to find the one that’s best friends with 8. It’s like trying to find the perfect flavor of ice cream – you just know when you’ve found the one that makes you do a little happy dance!
Let’s try a few, shall we? What about 800? Hmm, 800 divided by 8? Well, 8 goes into 8 once, and then we’ve got those two zeros. So, 800 is totally divisible by 8! That’s one possibility. But is it the only one? What if we try a different digit? Let's keep our eyes peeled and see if we can find another number that’s just as friendly with 8. Remember, we’re on a mission to find the digit.
Consider 808. Is 808 divisible by 8? Let's think. 800 is divisible by 8, and 8 is also divisible by 8 (that’s a super easy one!). So, 800 + 8 = 808. If both parts are divisible by 8, then their sum will be too! So, 808 is also divisible by 8. This is getting exciting! It feels like we’re unearthing hidden treasures!
Now, we’re getting closer, but we need to be absolutely sure. The rule for divisibility by 8 is actually a bit more specific than just any three digits. It’s about the last three digits. So, we’re looking at 380_. The critical part is the last three digits: 80_. We need to find a digit that, when it goes into that blank, makes the number 80_ divisible by 8. This is where the real magic happens!
Let’s think about multiples of 8. We’ve got 8, 16, 24, 32, 40, 48, 56, 64, 72, 80… Aha! We found 80! That means if the last two digits were 80, then the number would be divisible by 8. But we’re looking for 80_ to be divisible by 8. So, let’s think about multiples of 8 that start with 80. We already found 80 itself. What comes after 80 if we're looking at numbers that 8 can divide perfectly? Let’s keep going. 80 + 8 = 88. Is 88 divisible by 8? Yes, 8 times 11 is 88! So, if our last three digits were 808, that would be divisible by 8. And that's exactly the pattern we've been seeing!
So, let's put it all together. We have 380_. We need the last three digits, 80_, to be divisible by 8. We discovered that 808 is divisible by 8. This means that if we replace the underscore with the digit 8, our number becomes 3808. And because 808 is perfectly divisible by 8, the whole number 3808 is also perfectly divisible by 8! It’s like the missing piece was hiding in plain sight, just waiting for its moment to shine. You found it! You are a number detective extraordinaire! High fives all around!"