Find The Sum Of All Natural Divisors Of 24

Hey guys! Ever found yourself scratching your head over a math problem that seems simple but has a sneaky twist? Today, we're going to tackle one such problem: finding the sum of all natural divisors of the number 24. It’s a classic math teaser, and we'll break it down step-by-step so you can ace it every time. So, let's dive in and make math a bit more fun!

Understanding Divisors

Before we jump into the specifics of 24, let's get our heads around what divisors actually are. In the world of numbers, a divisor is a whole number that divides another number evenly, leaving no remainder. Think of it like sharing cookies: if you have 12 cookies and 3 friends, each friend can get 4 cookies with none left over. So, 3 is a divisor of 12.

To really nail this concept, let's consider a few examples. For the number 10, the divisors are 1, 2, 5, and 10. Why? Because 10 ÷ 1 = 10, 10 ÷ 2 = 5, 10 ÷ 5 = 2, and 10 ÷ 10 = 1 – all nice, clean divisions. Similarly, for the number 16, the divisors are 1, 2, 4, 8, and 16. See the pattern? We're looking for numbers that fit perfectly into the number we're examining.

Understanding this fundamental concept of divisors is crucial because it forms the bedrock for many other mathematical concepts. From simplifying fractions to understanding prime factorization, divisors play a pivotal role. Plus, when you encounter more complex problems, having a solid grasp of divisors can make the process much smoother. It's like having the right tool in your toolbox – it just makes the job easier! So, remember, divisors are your friends in the math world, always ready to help you break down numbers into manageable parts. Now that we've got this down, let's move on to our main challenge: figuring out the divisors of 24.

Finding the Natural Divisors of 24

Alright, now that we're all comfy with the idea of divisors, let's zoom in on our main number: 24. Our mission is to hunt down all the natural divisors of 24. Natural numbers, just to refresh, are the positive whole numbers we use for counting (1, 2, 3, and so on). No fractions, no decimals – just good ol' whole numbers. So, we're looking for all the whole numbers that can divide 24 without leaving any remainders.

To make sure we don't miss any, we'll go through the numbers one by one, starting with 1. Does 1 divide 24 evenly? Absolutely! 24 ÷ 1 = 24, so 1 is a divisor. Next up, 2. Does 2 fit into 24 nicely? You bet! 24 ÷ 2 = 12, so 2 is also a divisor. Let's keep going. 3 works too, since 24 ÷ 3 = 8. And what about 4? Yep, 24 ÷ 4 = 6, so 4 is in our divisor squad.

Now, let's try 5. Does 5 divide 24 evenly? Nope, we get a remainder, so 5 is out. Next, we have 6. Ah, we already found that 24 ÷ 6 = 4, so 6 is indeed a divisor. Notice something interesting here? We're starting to see pairs of divisors (like 4 and 6) that multiply to give 24. This can be a handy shortcut! Now, let's consider 7. Nope, 7 doesn't divide 24 evenly. How about 8? Well, 24 ÷ 8 = 3, so 8 is a divisor. And look, we've found another pair – 3 and 8.

We're on the home stretch now. Let's check 12. 24 ÷ 12 = 2, so 12 is a divisor. And guess what? We've got another pair – 2 and 12. The last number we need to check is 24 itself. And yes, 24 ÷ 24 = 1, so 24 is also a divisor. We've now completed our divisor treasure hunt! So, the natural divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Pat yourself on the back – that's the trickiest part done! Now, we just need to add them all up.

Calculating the Sum of Divisors

Okay, team, we've successfully identified all the natural divisors of 24. That was quite the detective work, wasn't it? Now comes the part where we put those divisors to work and calculate their sum. This is where our basic addition skills come into play, but don't worry, it's as straightforward as it gets. We're simply going to add up all the numbers we found to be divisors of 24.

So, let's line them up: 1, 2, 3, 4, 6, 8, 12, and 24. Now, we add them together: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24. To make things easier, we can pair up numbers that add up nicely. For instance, 1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, and then we have 4 + 6 = 10. So, our sum now looks like this: 25 + 14 + 11 + 10.

Let's keep adding! 25 + 14 gives us 39, and 11 + 10 equals 21. So, now we have 39 + 21. And finally, 39 + 21 equals 60. Voila! We've cracked it. The sum of all the natural divisors of 24 is 60. See, it wasn't so daunting after all, was it? We took it step by step, and now we have our answer.

This calculation is more than just a math problem; it's a method we can use for any number. Whether it's a small number like 15 or a larger one like 100, the process remains the same: identify the divisors and then add them up. It's a valuable skill to have in your math toolkit, and it's something you'll encounter in various areas of mathematics. So, remember this process, practice it with different numbers, and you'll become a divisor-summing pro in no time! Now that we've got our sum, let's circle back to the original question and nail down the correct answer.

Identifying the Correct Answer

Alright, mathletes, we've done the hard yards! We've found all the natural divisors of 24 and calculated their sum. We know that the sum is 60. Now, it's time to put on our detective hats one last time and match our answer with the options provided. Remember the options? They were A) 12, B) 36, C) 60, and D) 72.

We've got our answer, which is 60, clear as day. So, let's scan through the options and see which one matches. Option A is 12 – nope, that's not our sum. Option B is 36 – close, but no cigar. Option C is 60 – bingo! That's the one. And just to be sure, let's check Option D, which is 72 – definitely not our answer.

So, the correct answer is C) 60. We've successfully navigated through the problem, found the sum of the divisors, and matched it to the correct option. Give yourselves a round of applause, guys! You've shown some serious math prowess. This is a classic example of how breaking down a problem into smaller, manageable steps can make even the trickiest questions seem much easier.

This skill of problem-solving is super valuable, not just in math but in all areas of life. When faced with a challenge, remember our approach: understand the question, break it down, solve each part, and then put it all together to find the final answer. It's a strategy that works wonders, and it's something you can apply to all sorts of situations. So, keep practicing, keep exploring, and keep those math muscles strong! Now that we've nailed this problem, let's recap the key steps we took to get there.

Recap and Key Takeaways

Fantastic job, everyone! We've successfully navigated the challenge of finding the sum of the natural divisors of 24. We not only arrived at the correct answer but also reinforced some crucial math concepts along the way. So, let's take a moment to recap the journey we've been on and highlight the key takeaways. This will help solidify our understanding and make sure we're ready to tackle similar problems in the future.

First, we started by defining what divisors are. Remember, a divisor is a number that divides another number evenly, leaving no remainder. We looked at examples like the divisors of 10 and 16 to get a solid grasp of this concept. Understanding divisors is fundamental, as it's a building block for many other mathematical operations and concepts.

Next, we zoomed in on the number 24 and embarked on a divisor treasure hunt. We systematically checked each number to see if it divided 24 evenly. We discovered that the natural divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. This step required a bit of patience and careful checking, but we got there by breaking down the task and taking it one number at a time.

Once we had our list of divisors, we moved on to the calculation phase. We added up all the divisors: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24. We even used a handy trick of pairing numbers to make the addition easier. This led us to the sum of 60, which is the answer we were looking for.

Finally, we matched our answer with the options provided and confidently selected C) 60 as the correct choice. We celebrated our success and reflected on the problem-solving approach we used. This included understanding the question, breaking it down into steps, solving each part, and then combining the results to find the final answer.

So, what are the key takeaways from this adventure? Firstly, understanding the definition of divisors is crucial. Secondly, a systematic approach is essential for finding all the divisors of a number. Thirdly, breaking down a problem into smaller steps makes it more manageable. And fourthly, always double-check your work and match your answer with the options provided. These are valuable lessons that will serve you well in your math journey and beyond. Keep practicing, keep exploring, and remember, math can be fun!

So, what's the sum of all the natural divisors of the number 24?

The answer is C) 60.