LCM Of 2*3*5*7 And 2*3*7: Step-by-Step Solution

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Hey guys! Let's dive into how to find the Least Common Multiple (LCM) of two numbers, specifically 2 * 3 * 5 * 7 and 2 * 3 * 7. This is a common problem in math, and understanding the process will help you tackle similar questions with confidence. We'll break it down step-by-step, so you know exactly how to solve it and how to write it down in your notebook. So, let’s get started!

Understanding the Least Common Multiple (LCM)

Before we jump into solving the problem, let's quickly recap what the Least Common Multiple (LCM) actually means. In simple terms, the LCM of two or more numbers is the smallest number that is a multiple of each of those numbers. Think of it as the smallest number that all the given numbers can divide into evenly.

Why is this important? Well, the LCM is super useful in various areas of mathematics, especially when you're dealing with fractions, adding and subtracting them, or solving problems involving ratios and proportions. It helps you find a common denominator, which makes calculations much easier. So, mastering how to find the LCM is a fundamental skill. Now that we're all on the same page about what LCM is, let’s move on to the methods we can use to calculate it, focusing on the prime factorization method, which is perfect for problems like the one we have.

Methods to Calculate the LCM

There are a few different ways to calculate the LCM, but we're going to focus on the prime factorization method, as it's particularly effective for larger numbers and expressions like the ones we have. Other methods include listing multiples, but this can become quite cumbersome when the numbers get bigger. Prime factorization, on the other hand, provides a systematic approach that ensures accuracy and efficiency. We’ll briefly touch on why prime factorization is so effective, and then we'll dive into the specific steps.

Why Prime Factorization?

Prime factorization breaks down each number into its prime factors – the prime numbers that multiply together to give the original number. This method is powerful because it allows us to identify all the unique prime factors present in each number. By taking the highest power of each prime factor that appears in any of the numbers, we can construct the LCM. This ensures that the LCM is divisible by each of the original numbers, and it's the smallest such number.

Now, let’s get into the step-by-step process of using the prime factorization method. We'll apply it to our specific problem, so you'll see exactly how it works in practice. This will not only help you solve this problem but also equip you with a skill you can use in many other mathematical contexts.

Step 1: Prime Factorization of the Given Numbers

Okay, let's start with the first crucial step: prime factorization. This means we need to break down each number into its prime factors. Remember, prime factors are prime numbers that multiply together to give you the original number. In our case, we have two numbers:

  1. 2 * 3 * 5 * 7
  2. 2 * 3 * 7

Lucky for us, these numbers are already conveniently expressed in their prime factorized form! This makes our job a whole lot easier. We can clearly see the prime factors involved. For the first number, the prime factors are 2, 3, 5, and 7. For the second number, they are 2, 3, and 7. Why is this important? Because now we can easily identify which prime factors we need to include in our LCM calculation.

This step is the foundation of the entire process. If you were starting with composite numbers (numbers that are not prime), you would need to break them down into their prime factors first. There are several methods to do this, such as using factor trees or division. But in this case, we've been given a head start, which is great! Now that we have the prime factorizations, we can move on to the next step, where we'll identify the highest powers of each prime factor.

Step 2: Identify the Highest Powers of Each Prime Factor

Now that we have our numbers in their prime factorized form, the next step is to identify the highest powers of each prime factor present in either number. This is a crucial step because the LCM must be divisible by both numbers, so it needs to include all the prime factors raised to their highest powers.

Let's take a look at our prime factors again:

  1. 2 * 3 * 5 * 7
  2. 2 * 3 * 7

We can see the prime factors involved are 2, 3, 5, and 7. Now, let's examine each prime factor individually:

  • 2: The highest power of 2 that appears in either number is 2¹ (which is just 2). It appears once in both numbers.
  • 3: Similarly, the highest power of 3 is 3¹ (or 3), appearing once in both numbers.
  • 5: The highest power of 5 is 5¹ (or 5). It only appears in the first number.
  • 7: The highest power of 7 is 7¹ (or 7), appearing once in both numbers.

So, we've identified the highest powers of each prime factor. Why is this step so important? Because these highest powers are the building blocks of our LCM. We need to make sure our LCM includes each of these prime factors raised to their respective highest powers to ensure it's divisible by both original numbers. In the next step, we'll use these highest powers to actually calculate the LCM.

Step 3: Calculate the LCM

Alright, we've done the groundwork – we've got our prime factorizations and we've identified the highest powers of each prime factor. Now comes the exciting part: calculating the LCM. This is where it all comes together. Remember, the LCM is the product of all the highest powers of the prime factors we identified in the previous step.

We found that the highest powers of the prime factors are:

  • 2¹ (or 2)
  • 3¹ (or 3)
  • 5¹ (or 5)
  • 7¹ (or 7)

So, to calculate the LCM, we simply multiply these together:

LCM = 2 * 3 * 5 * 7

Now, let's do the multiplication:

LCM = 2 * 3 = 6 LCM = 6 * 5 = 30 LCM = 30 * 7 = 210

So, the LCM of 2 * 3 * 5 * 7 and 2 * 3 * 7 is 210. How cool is that? We've successfully found the smallest number that both 2 * 3 * 5 * 7 and 2 * 3 * 7 divide into evenly. This is a really important result, and you can use it in all sorts of mathematical problems. Next, we'll discuss how to write this solution down neatly in your notebook, so you can show your work clearly and get full credit!

Step 4: How to Write it Down in Your Notebook

Okay, you've solved the problem – awesome! But just as important as getting the right answer is knowing how to present your solution clearly and logically in your notebook. This not only helps your teacher understand your thought process but also helps you review your work later on. Let's break down how to write down the solution for this LCM problem step-by-step:

  1. State the Problem:
    • Start by clearly stating the problem you're solving. This helps provide context. You can write something like:

      "Find the Least Common Multiple (LCM) of 2 * 3 * 5 * 7 and 2 * 3 * 7."

  2. Prime Factorization:
    • Show the prime factorization of each number. In this case, they're already given, so you can simply write:

      "Prime factorization:

      • 2 * 3 * 5 * 7
      • 2 * 3 * 7"
  3. Identify Highest Powers:
    • Clearly list the highest power of each prime factor:

      "Highest powers of prime factors:

      • 2¹
      • 3¹
      • 5¹
      • 7¹"
  4. Calculate the LCM:
    • Write out the multiplication to calculate the LCM:

      "LCM = 2 * 3 * 5 * 7"

    • Then, show the final result:

      "LCM = 210"

  5. Final Answer:
    • Clearly state your final answer. This makes it easy for the reader (and you!) to quickly identify the solution:

      "Therefore, the LCM of 2 * 3 * 5 * 7 and 2 * 3 * 7 is 210."

By following these steps, you’ll not only get the correct answer but also present your work in a clear, organized, and easy-to-understand manner. Why is this important? Clear communication is a key skill in mathematics and in life! So, practice writing your solutions in a structured way, and it will become second nature. Remember, showing your work is just as important as getting the answer right!

Conclusion

So, there you have it, guys! We've walked through the entire process of finding the Least Common Multiple (LCM) of 2 * 3 * 5 * 7 and 2 * 3 * 7. We started by understanding what LCM means, then we dived into the prime factorization method, identified the highest powers of each prime factor, calculated the LCM, and finally, we learned how to present the solution clearly in your notebook.

Remember, practice makes perfect! The more you work through problems like this, the more confident you’ll become. The LCM is a fundamental concept in math, and mastering it will help you in various areas, from fractions to algebra and beyond. So, keep practicing, keep exploring, and most importantly, keep having fun with math!

If you have any questions or want to tackle more LCM problems, feel free to ask. And until next time, happy calculating!