Smallest Number With Remainder 4: Divisible By 12, 15, 10

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Hey guys! Have you ever stumbled upon a math problem that seems a bit tricky at first glance? Well, let's dive into one today. We're going to figure out the smallest number that leaves a remainder of 4 when divided by 12, 15, and 10. Sounds like a puzzle, right? Don't worry; we'll break it down step by step and make it super easy to understand.

Understanding the Problem

Okay, so let's rephrase the problem to make sure we're all on the same page. We're searching for a number that, when you divide it by 12, 15, or 10, always has 4 left over. That remainder of 4 is the key here. Think of it like this: if the remainder wasn't there, our number would be perfectly divisible by 12, 15, and 10. So, what do we need to find first? We need to find the smallest number that is perfectly divisible by all three. This is where the concept of the Least Common Multiple, or LCM, comes into play.

What is the Least Common Multiple (LCM)?

The Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more numbers. It’s a fundamental concept in number theory and is super useful in solving problems like the one we have. Imagine you have several gears meshing together; the LCM helps you figure out when they will all align again. In our case, it helps us find the smallest number that 12, 15, and 10 can all divide into evenly. Understanding LCM is crucial because it forms the foundation for solving our main problem. Without it, we'd be shooting in the dark, trying out random numbers and hoping for the best. So, let's get into how we actually calculate the LCM.

Finding the Least Common Multiple (LCM)

Alright, let's roll up our sleeves and find the LCM of 12, 15, and 10. There are a couple of ways we can do this, but one of the most common and straightforward methods is the prime factorization method. This method involves breaking down each number into its prime factors – those prime numbers that, when multiplied together, give you the original number. Trust me; it's not as intimidating as it sounds! Once we have the prime factors, we can easily piece together the LCM.

Prime Factorization Method

So, how does this prime factorization method work? First, we express each number as a product of its prime factors. Let's start with 12. We can break it down as 2 x 2 x 3, or 2² x 3. Next, let's tackle 15. That's simply 3 x 5. And finally, 10 breaks down into 2 x 5. Now we have the building blocks for each number. To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations and multiply them together. Let’s walk through this step by step to make it crystal clear.

  1. List the Prime Factors:

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

  2. Identify Highest Powers:

    • The highest power of 2 is 2² (from 12).
    • The highest power of 3 is 3¹ (appears in 12 and 15).
    • The highest power of 5 is 5¹ (appears in 15 and 10).

  3. Multiply Highest Powers:

    • LCM (12, 15, 10) = 2² x 3 x 5 = 4 x 3 x 5 = 60

So, the LCM of 12, 15, and 10 is 60. This means 60 is the smallest number that all three of these numbers can divide into without leaving a remainder. But remember, our original problem has that pesky remainder of 4 to deal with. We’re not quite finished yet!

Accounting for the Remainder

Okay, we've figured out that 60 is the smallest number divisible by 12, 15, and 10. But our question asks for the smallest number that leaves a remainder of 4 when divided by these numbers. So, what's the next step? It's actually pretty straightforward. Since we want a remainder of 4, all we need to do is add 4 to our LCM. This will give us a number that, when divided by 12, 15, or 10, will always have 4 left over.

Adding the Remainder to the LCM

To find the number we're looking for, we simply add the remainder (4) to the LCM (60):Number = LCM + RemainderNumber = 60 + 4Number = 64So, 64 is the smallest number that leaves a remainder of 4 when divided by 12, 15, and 10. We’ve got our answer! But let’s just double-check to make sure everything adds up correctly. It’s always a good idea to verify your solution, especially in math problems. We want to be 100% confident in our result.

Verification

To verify our answer, we'll divide 64 by 12, 15, and 10 and see if we indeed get a remainder of 4 in each case. This is a crucial step because it ensures we haven't made any mistakes along the way. Math can be tricky, and it's easy to slip up on a calculation, so verification is our safety net. Let's get started!

Checking the Answer

Let's run the divisions:

  • 64 ÷ 12 = 5 with a remainder of 4 (12 x 5 = 60, and 64 - 60 = 4)
  • 64 ÷ 15 = 4 with a remainder of 4 (15 x 4 = 60, and 64 - 60 = 4)
  • 64 ÷ 10 = 6 with a remainder of 4 (10 x 6 = 60, and 64 - 60 = 4)

Great! In all three cases, we get a remainder of 4. This confirms that our answer is correct. We found the smallest number that satisfies the given conditions. It’s always satisfying when a plan comes together, right? We've successfully solved the problem by breaking it down into manageable steps and verifying our solution. Now, let’s recap the entire process to solidify our understanding.

Conclusion

Awesome! We've successfully found the smallest number that leaves a remainder of 4 when divided by 12, 15, and 10. It's 64! We started by understanding the problem, then found the Least Common Multiple (LCM) of 12, 15, and 10, which was 60. Finally, we added the remainder 4 to the LCM, giving us our answer. And of course, we verified our answer to make sure everything was spot on. You guys nailed it!

Recap of Steps

Let's quickly recap the steps we took to solve this problem:

  1. Understand the Problem: We identified that we needed to find a number that leaves a remainder of 4 when divided by 12, 15, and 10.
  2. Find the LCM: We determined the Least Common Multiple (LCM) of 12, 15, and 10 using the prime factorization method. The LCM was 60.
  3. Account for the Remainder: We added the remainder (4) to the LCM (60) to find the required number.
  4. Verify the Answer: We checked our solution by dividing the result (64) by 12, 15, and 10 to ensure we got a remainder of 4 in each case.

This kind of problem might seem tough at first, but by breaking it down into smaller, manageable steps, it becomes much easier. Remember, the key is to understand the underlying concepts, like LCM, and then apply them methodically. Keep practicing, and you'll become a math whiz in no time! Keep an eye out for more fun math problems, and let's tackle them together. You've got this!