Finding The Smallest Number Of Books With Remainder 8

by Scholario Team 54 views

Hey guys! Today, we're diving into a fun math problem that involves finding the smallest number of books that leaves a remainder of 8 when divided into boxes of different sizes. This is a classic problem that combines the concepts of the least common multiple (LCM) and remainders. Let's break it down step by step so you can totally ace similar problems in the future!

Understanding the Problem

So, the problem goes something like this: A school received a certain number of books as a gift. If they try to pack these books into boxes of 10, 12, or 15 books each, there are always 8 books left over. The big question is, what's the smallest possible number of books the school received? This isn't just some abstract math problem, guys. These kinds of questions pop up in real-world situations all the time, from inventory management to scheduling. When you get the hang of solving them, you’re not just learning math, you’re boosting your problem-solving skills in general. Think of it as mental gymnastics that makes you sharper in all areas of life!

To really get our heads around it, let's pull out the key pieces of information. We know we're looking for the smallest number, which means we're dealing with a minimum value. We also know that no matter if we divide the books into groups of 10, 12, or 15, we always have 8 books sitting there, not fitting into any group. That remainder of 8 is super important because it tells us the number we're looking for is 8 more than a number that can be divided evenly by 10, 12, and 15. This is where the least common multiple comes into play, as it helps us find that evenly divisible number.

Before we jump into crunching numbers, let's take a step back and think about why this works. Imagine you have a pile of LEGO bricks. You try to make towers of 10, 12, and 15, but you always have 8 bricks left over. To find the smallest number of bricks you could have, you first need to find the smallest number of bricks that can make complete towers of 10, 12, and 15. Then, you just add those extra 8 bricks back in. This hands-on way of thinking about the problem can make it way easier to grasp, especially when the numbers start to get bigger or the problem gets more complicated.

Finding the Least Common Multiple (LCM)

The first key step in solving this problem is to find the least common multiple (LCM) of 10, 12, and 15. The LCM is the smallest number that is a multiple of each of these numbers. This is crucial because it represents the smallest number of books that could be perfectly divided into boxes of 10, 12, or 15 with no remainders. There are a couple of ways we can find the LCM. One method is listing out the multiples of each number until we find a common one. Another more efficient method is using prime factorization.

Let’s start with the listing method. We list the multiples of each number:

  • Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …
  • Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, …
  • Multiples of 15: 15, 30, 45, 60, 75, 90, 105, …

Looking at these lists, we can see that the smallest number that appears in all three lists is 60. So, the LCM of 10, 12, and 15 is 60. This means 60 is the smallest number of books that can be packed perfectly into boxes of 10, 12, or 15. Pretty cool, right? But, we're not done yet, because we still need to account for those extra 8 books.

Now, let’s try the prime factorization method, which is super handy when dealing with larger numbers. Prime factorization is breaking down each number into its prime factors – those prime numbers that, when multiplied together, give you the original number. Here's how it works:

  • 10 = 2 x 5
  • 12 = 2 x 2 x 3 = 2² x 3
  • 15 = 3 x 5

To find the LCM using prime factors, we take the highest power of each prime factor that appears in any of the factorizations and multiply them together. In this case, the prime factors are 2, 3, and 5. The highest power of 2 is 2² (from the factorization of 12), the highest power of 3 is 3 (appears in both 12 and 15), and the highest power of 5 is 5 (appears in both 10 and 15). So, the LCM is 2² x 3 x 5 = 4 x 3 x 5 = 60. Ta-da! We get the same answer using both methods. Whether you prefer listing multiples or prime factorization, the key is to find the LCM accurately. It’s the foundation for solving this whole problem.

Adding the Remainder

Okay, we've nailed the least common multiple. We know that 60 is the smallest number of books that can be divided evenly into boxes of 10, 12, or 15. But remember, the problem stated that there are always 8 books left over. So, what do we do with that information? We simply add the remainder to the LCM!

This step is actually quite intuitive when you think about it. The LCM gives us the base number that satisfies the divisibility condition (divisible by 10, 12, and 15). The remainder represents the extra books that couldn't fit into those boxes. To find the total number of books, we need to include both the books that fit perfectly and the leftovers. It’s like saying, “Okay, we can fill all these boxes, but we still have these 8 sitting here.”

So, we take our LCM, which is 60, and add the remainder, which is 8:

60 + 8 = 68

And there we have it! The smallest number of books the school could have received is 68. Let's just take a moment to appreciate how elegantly this works. By finding the LCM and adding the remainder, we've solved a seemingly tricky problem with just a couple of simple operations. This is the beauty of math – complex problems can often be broken down into smaller, more manageable steps.

To really solidify this, let’s think about it in terms of our earlier LEGO brick analogy. We figured out that 60 bricks would make complete towers of 10, 12, or 15. But we also knew we had 8 extra bricks. So, the total number of bricks we started with must have been 60 (the towers) plus 8 (the extras), giving us 68 bricks in total. See how it all clicks together?

Verifying the Solution

It's always a good idea to verify our solution to make sure we didn't make any silly mistakes along the way. How can we check if 68 is indeed the correct answer? We can divide 68 by 10, 12, and 15 and see if we get a remainder of 8 in each case. This is like a mini-test to make sure our answer holds up under scrutiny. Think of it as double-checking your work before handing in an assignment – it can save you from careless errors and boost your confidence in your solution.

Let's do the divisions:

  • 68 ÷ 10 = 6 with a remainder of 8 (Check!)
  • 68 ÷ 12 = 5 with a remainder of 8 (Check!)
  • 68 ÷ 15 = 4 with a remainder of 8 (Check!)

Awesome! Our solution checks out. When we divide 68 by 10, 12, or 15, we consistently get a remainder of 8. This confirms that 68 is indeed the smallest number of books that satisfies the conditions of the problem. Verifying your answer is a crucial step in problem-solving, especially in math. It’s not just about getting an answer; it’s about knowing that your answer is correct and understanding why. This process builds a deeper understanding of the concepts involved and helps you avoid common pitfalls.

Moreover, verifying solutions reinforces the connection between the abstract math and the real-world scenario. In this case, it's about making sure our calculated number of books actually makes sense in the context of packing books into boxes. This kind of thinking is what makes math useful and relevant in everyday life. When you can see how the math works in a practical situation, it becomes less like a set of rules and more like a powerful tool for solving problems.

Conclusion

So, the smallest number of books that were given to the school is 68. We solved this by first finding the least common multiple of 10, 12, and 15, which was 60, and then adding the remainder of 8. This type of problem highlights the importance of understanding LCM and how it can be applied in various situations.

This wasn't just about finding the right number, guys; it was about the journey we took to get there. We broke down the problem, identified the key information, used the least common multiple, and then incorporated the remainder. We even took the extra step of verifying our answer, which is super important in math and in life. Problem-solving is a skill, and like any skill, it gets better with practice. So, the more you tackle these kinds of challenges, the more confident and capable you'll become.

Math isn’t just a subject you study in school; it’s a way of thinking. It's about looking at a situation, identifying patterns, and finding logical solutions. The skills we used in this problem – breaking things down, finding commonalities, and double-checking our work – are skills that will serve you well in all sorts of areas, from managing your finances to planning a project at work. So, keep those math muscles flexed, and remember that every problem is an opportunity to learn and grow!

Keep practicing similar problems, and you'll become a pro at solving them in no time! You got this! Remember, the world is full of problems waiting to be solved, and you have the tools to tackle them. So, go out there and put your math skills to the test. And hey, if you ever get stuck, don't hesitate to ask for help. Math is a team sport, too! Learning together and sharing insights can make the whole process even more rewarding. Now, go conquer those numbers!