Divisibility Of 24,816: Factors And Sum Of Digits

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Hey guys! Ever wondered how to figure out what numbers can perfectly divide a big number like 24,816? It's actually a fun math puzzle, and we're going to break it down step by step. We'll look at some cool tricks and rules that'll help you become a divisibility whiz. So, let's dive in and explore the fascinating world of numbers and their factors!

Understanding the Problem

The question we're tackling is: Given that the number 24,816 ends with an even digit and the sum of its digits is 21, which numbers divide it exactly? This is a classic divisibility problem that combines a couple of different concepts. First, we have the rule for even numbers, and second, we have the rule for the sum of digits. By combining these, we can narrow down the possible divisors.

Let's start by dissecting the number 24,816 itself. We know it's an even number because it ends in 6. This is our first clue! Even numbers are always divisible by 2. Next, the problem tells us that the sum of its digits is 21. So, 2 + 4 + 8 + 1 + 6 = 21. This piece of information is crucial because it links to another divisibility rule. We'll see how this helps us figure out other factors shortly.

The question is asking us to identify which numbers from a given set divide 24,816 without leaving a remainder. This means we're looking for factors of 24,816. To solve this, we will use divisibility rules as our superpower, making the process much easier than just trying to divide 24,816 by every possible number. Think of these rules as shortcuts to finding the right answers. So, buckle up, and let’s get started!

Divisibility Rules: Our Superpowers

Okay, so what are these magical divisibility rules we keep talking about? Well, they're not magic, but they sure feel like it sometimes! Divisibility rules are basically shortcuts that help us determine if a number is divisible by another number without actually doing long division. They're super handy for problems like this one.

Let's start with the basics:

  • Divisibility by 2: This is the easiest one! A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). We already know 24,816 passes this test because it ends in 6.
  • Divisibility by 3: Here's where the sum of digits comes in. A number is divisible by 3 if the sum of its digits is divisible by 3. Remember, the sum of the digits of 24,816 is 21. Is 21 divisible by 3? You bet! 21 / 3 = 7, so 24,816 is divisible by 3.
  • Divisibility by 4: For this one, we look at the last two digits of the number. If the number formed by the last two digits is divisible by 4, then the whole number is divisible by 4. In our case, the last two digits are 16. Since 16 is divisible by 4 (16 / 4 = 4), 24,816 is also divisible by 4.
  • Divisibility by 5: This is another easy one. A number is divisible by 5 if its last digit is either 0 or 5. 24,816 ends in 6, so it's not divisible by 5.
  • Divisibility by 6: To be divisible by 6, a number must be divisible by both 2 and 3. We already know 24,816 is divisible by both 2 and 3, so it's also divisible by 6.
  • Divisibility by 8: Similar to the rule for 4, but this time we look at the last three digits. If the number formed by the last three digits is divisible by 8, then the whole number is divisible by 8. The last three digits of 24,816 are 816. Is 816 divisible by 8? Yes, 816 / 8 = 102, so 24,816 is divisible by 8.
  • Divisibility by 9: This is similar to the rule for 3. A number is divisible by 9 if the sum of its digits is divisible by 9. We know the sum of the digits is 21. Is 21 divisible by 9? Nope! So, 24,816 is not divisible by 9.

These divisibility rules are super useful, and with a little practice, you'll be able to spot factors in no time! Now, let’s apply these rules to our number, 24,816, and see what we can find out.

Applying the Rules to 24,816

Alright, we've got our divisibility rule toolkit ready, so let's put them to work on 24,816. Remember, the goal is to figure out which numbers divide 24,816 exactly.

We already know a few things:

  • Divisible by 2: Yes, because it ends in 6 (an even number).
  • Divisible by 3: Yes, because the sum of its digits (21) is divisible by 3.

Let’s go through the other rules:

  • Divisible by 4: The last two digits are 16, which is divisible by 4. So, yes!
  • Divisible by 5: No, because it doesn't end in 0 or 5.
  • Divisible by 6: Yes, because it's divisible by both 2 and 3.
  • Divisible by 8: The last three digits are 816, which is divisible by 8. So, yes!
  • Divisible by 9: No, because the sum of its digits (21) is not divisible by 9.

So, just by applying these rules, we've figured out that 24,816 is divisible by 2, 3, 4, 6, and 8. How cool is that? We've narrowed down the possibilities without even doing any long division. This is the power of divisibility rules in action!

Determining the Exact Divisors

Now that we've used our divisibility rules to identify some potential divisors, let's bring it all together. The question asked us which numbers divide 24,816 exactly, given that it ends with an even digit and the sum of its digits is 21. Based on our analysis, we know the following:

  • 24,816 is divisible by 2 because it ends in an even digit.
  • 24,816 is divisible by 3 because the sum of its digits (21) is divisible by 3.
  • 24,816 is divisible by 4 because its last two digits (16) are divisible by 4.
  • 24,816 is divisible by 6 because it is divisible by both 2 and 3.
  • 24,816 is divisible by 8 because its last three digits (816) are divisible by 8.
  • 24,816 is not divisible by 5 because it does not end in 0 or 5.
  • 24,816 is not divisible by 9 because the sum of its digits (21) is not divisible by 9.

So, from the options provided (which we assume included 2, 3, 4, 5, 6, 8, and 9), the numbers that divide 24,816 exactly are 2, 3, 4, 6, and 8. You see, guys? Using divisibility rules makes these kinds of problems much more manageable!

Conclusion: The Magic of Divisibility

Wow, we've really taken a deep dive into the world of divisibility, haven't we? We started with a question about the number 24,816 and its factors, and we ended up exploring a whole set of cool math shortcuts. The key takeaway here is that divisibility rules are powerful tools that can help you quickly determine if a number is divisible by another number without getting bogged down in long division. Think of them as your secret weapon for tackling number problems!

We saw how the rules for 2, 3, 4, 5, 6, 8, and 9 work, and we applied them to 24,816. By looking at the last digit, the sum of the digits, and the last two or three digits, we were able to identify several factors of 24,816. It's like being a math detective, finding clues and solving the mystery of which numbers fit perfectly into 24,816.

Remember, the more you practice these rules, the faster and more confident you'll become in using them. So, next time you encounter a divisibility problem, don't panic! Just pull out your divisibility rule toolkit and start sleuthing. Math can be fun, especially when you have the right tools and tricks up your sleeve. Keep exploring, keep learning, and keep those numbers crunching! You've got this!