Divisibility By 4: Finding The Missing Digit

Hey everyone! Let's dive into a fun math problem today that involves figuring out divisibility rules. Specifically, we're going to tackle a question about finding a missing digit in a four-digit number so that the whole number is divisible by 4. If you love puzzles and enjoy playing with numbers, this one's for you! We'll break it down step by step, so you'll not only get the answer but also understand the logic behind it. So, let's get started and unlock the mystery of this missing digit!

Understanding the Question

Okay, so the question we're tackling is: "For the four-digit number 724lacksquare with distinct digits, how many different digits can be written in place of $\blacksquare$ so that the number is divisible by 4?" Let's break this down a bit. The key here is divisibility by 4, and we need to find how many possible digits can replace the square while ensuring the entire number remains divisible by 4. This means we need to know the rule for divisibility by 4 and how to apply it. This isn't just about finding any number; we've got the added condition that all digits in the final number must be different. This little detail adds a layer of complexity, making the problem a bit more interesting. This part is super important because it means we can't just pick any digit that makes the number divisible by 4; we also have to make sure it hasn't already been used in the number (7, 2, or 4). It’s like a mini-puzzle within the main puzzle. So, before we even start plugging in numbers, we need to keep this restriction in mind.

Divisibility Rule for 4

Now, let's talk about the golden rule that will help us crack this problem: the divisibility rule for 4. This rule is actually quite simple and super handy. A number is divisible by 4 if its last two digits are divisible by 4. That's it! We don't need to look at the whole number; just the last two digits. Think of it like this: 100 is divisible by 4, so any multiple of 100 is also divisible by 4. That means we only need to worry about the remainder after dividing by 100, which is exactly what the last two digits represent. So, instead of trying to divide the whole four-digit number by 4, we can focus solely on the number formed by the last two digits, which in our case is 4lacksquare. This makes our task way easier and more manageable. Remember, this rule is your best friend when you're dealing with divisibility by 4. It turns a potentially complex problem into a much simpler one. To make sure we really understand it, let’s try a quick example. Is 1236 divisible by 4? Instead of dividing the whole thing, we just look at 36. Is 36 divisible by 4? Yes, it is (36 / 4 = 9). So, 1236 is also divisible by 4. See how easy that is?

Applying the Rule to Our Problem

Alright, guys, now that we've got the divisibility rule for 4 down, let's apply it to our problem. Remember, we have the number 724lacksquare, and we need to find the digits that can replace the square so that the last two digits (4lacksquare) are divisible by 4. This is where the fun begins! We need to think of all the two-digit numbers that start with 4 and are divisible by 4. So, let’s list them out: 40, 44, 48. Easy peasy, right? But hold on a second! We're not done yet. Remember that sneaky little condition about distinct digits? Yeah, we can't forget about that. Looking back at our original number, 724lacksquare, we already have the digit 4 in the hundreds place. This means we can't use 44 as a possibility because the digits wouldn't be distinct. So, we can cross 44 off our list. Now we're left with 40 and 48. This is where attention to detail really pays off. It’s so easy to get caught up in the divisibility rule and forget about the other conditions of the problem. This step is crucial because it narrows down our options and brings us closer to the correct answer. So always remember to double-check all the conditions before making your final decision.

Checking for Distinct Digits

Okay, so we've narrowed down our possibilities to 40 and 48. But before we jump to any conclusions, we need to make absolutely sure that the digits we're plugging in for the square result in a four-digit number with distinct digits. This is like the final boss level of our problem, the last hurdle to clear before we claim victory. Let’s take each possibility one by one. First, let's try replacing the square with 0. This gives us the number 7240. Are all the digits distinct? Yes! We have 7, 2, 4, and 0 – all different. So, 0 is a valid option. Now, let's try replacing the square with 8. This gives us the number 7248. Again, let's check for distinct digits. We have 7, 2, 4, and 8 – all different. So, 8 is also a valid option. See how we’re not just blindly accepting the numbers? We're putting them through a rigorous test to make sure they fit all the criteria. This is what it means to be a careful problem-solver. It’s not enough to find numbers that work; we need to make sure they work completely.

Finding the Solution

Alright, we've done the heavy lifting, guys! We've applied the divisibility rule, considered the distinct digits condition, and checked our possibilities. Now it's time for the grand finale: finding the solution! We found that the possible digits that can replace the square are 0 and 8. This means there are two different digits that we can write in place of the square to make the number 724lacksquare divisible by 4 while keeping all the digits distinct. So, the answer to our question, "How many different digits can be written in place of $\blacksquare$ so that the number is divisible by 4?" is 2! Wasn't that a fun journey? We didn't just find the answer; we explored the problem, understood the rules, and carefully checked our work. This is what math is all about – not just getting the right answer, but understanding the process and the logic behind it. We've tackled the problem step by step, making sure we didn’t miss any crucial details. From recalling the divisibility rule for 4 to remembering the condition about distinct digits, we've shown how important it is to be thorough and methodical in problem-solving.

So, the final answer is:

B) 2

Conclusion

Great job, everyone! We've successfully solved this divisibility puzzle. Remember, the key to tackling math problems is to break them down into smaller, manageable steps. Don't be afraid to explore different possibilities and always double-check your work. And most importantly, have fun with it! Math can be like a game, and each problem is a new challenge to conquer. We started with a four-digit number with a missing digit and a condition about divisibility by 4. By applying the divisibility rule for 4 and being mindful of the distinct digits requirement, we narrowed down the possibilities and arrived at the correct answer. This problem wasn't just about finding the right number; it was about developing our problem-solving skills, our attention to detail, and our ability to think logically. Keep practicing, keep exploring, and keep challenging yourselves. The world of math is full of fascinating puzzles just waiting to be solved!