LCM Of 16/63 And 35/48: Step-by-Step Guide

Hey guys! Ever wondered how to find the least common multiple (LCM) of fractions? It might sound a bit tricky, but trust me, it's totally doable once you break it down. In this guide, we're going to tackle the question: What is the least common multiple of 16/63 and 35/48? We'll go through each step in detail, so you'll not only get the answer but also understand the process. So, let's dive in and make math a little less mysterious!

The least common multiple (LCM) is a fundamental concept in mathematics, particularly when dealing with fractions. Understanding LCM is crucial for various operations, including adding, subtracting, and comparing fractions with different denominators. The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. This concept is widely used in everyday life, from scheduling tasks to dividing quantities. When dealing with fractions, finding the LCM involves a slightly different approach compared to whole numbers, but the underlying principle remains the same: to find the smallest common multiple. This foundational knowledge is essential for simplifying fractions and performing more complex mathematical operations involving fractions. Mastering the LCM of fractions not only enhances your mathematical skills but also provides a practical tool for solving real-world problems. So, let's break down the steps and make sure you've got a solid grasp of how to tackle these problems.

Breaking Down the Problem: 16/63 and 35/48

Before we jump into the solution, let's take a closer look at the fractions we're dealing with: 16/63 and 35/48. To find the LCM of fractions, we actually use a slightly different approach than when we're working with whole numbers. Instead of finding the LCM directly, we'll be finding the LCM of the numerators (the top numbers) and the greatest common divisor (GCD) of the denominators (the bottom numbers). Understanding this key difference is crucial for solving these types of problems accurately. The numerators here are 16 and 35, while the denominators are 63 and 48. We'll need to work with these numbers individually to get to our final answer. Don't worry, we'll go through each step together, making sure everything is clear. So, let's start by looking at the numerators and denominators separately and see how we can break them down further to find the LCM and GCD.

Understanding Numerators and Denominators

Let's quickly recap what numerators and denominators are, just to make sure we're all on the same page. The numerator is the top number in a fraction, and it tells you how many parts of a whole you have. For instance, in the fraction 16/63, 16 is the numerator. On the other hand, the denominator is the bottom number, and it shows the total number of equal parts that the whole is divided into. In our example, 63 is the denominator. When we're finding the LCM of fractions, we focus on both of these parts, but in different ways. We'll be finding the LCM of the numerators (16 and 35) and the greatest common divisor (GCD) of the denominators (63 and 48). This might sound like a lot of jargon, but don't worry! We'll break it down step by step. So, now that we've clarified the roles of numerators and denominators, let's move on to finding the LCM of the numerators.

Step 1: Finding the LCM of the Numerators (16 and 35)

Okay, let's get started with the first part of our problem: finding the LCM of the numerators, which are 16 and 35. To do this, we'll use the prime factorization method. This involves breaking down each number into its prime factors – those prime numbers that multiply together to give you the original number. First, let's find the prime factors of 16. We can express 16 as 2 x 2 x 2 x 2, or 2^4. Now, let's do the same for 35. The prime factors of 35 are 5 and 7, since 35 = 5 x 7. Once we have the prime factorizations, finding the LCM is pretty straightforward. We take the highest power of each prime factor that appears in either factorization and multiply them together. This gives us the smallest number that both 16 and 35 can divide into evenly. So, let's see how this plays out in our calculation.

Prime Factorization of 16

The prime factorization of a number is like its unique fingerprint – it’s the set of prime numbers that, when multiplied together, give you that number. For 16, we want to find all the prime numbers that multiply to 16. We can start by dividing 16 by the smallest prime number, which is 2. 16 divided by 2 is 8. Now, we divide 8 by 2 again, which gives us 4. Divide 4 by 2, and we get 2. And finally, 2 divided by 2 is 1. So, we’ve broken 16 down to 2 × 2 × 2 × 2. That means 16 can be expressed as 2 to the power of 4, or 2^4. This prime factorization is super helpful because it tells us all the prime building blocks that make up 16. Knowing this, we can move on to finding the prime factorization of 35 and then use these prime factors to find the LCM.

Prime Factorization of 35

Now, let's break down 35 into its prime factors. Remember, we're looking for prime numbers that multiply together to give us 35. We can start by trying the smallest prime number, 2, but 35 isn’t divisible by 2. The next prime number is 3, but 35 isn’t divisible by 3 either. Let’s try 5. 35 divided by 5 is 7. And guess what? Both 5 and 7 are prime numbers! So, the prime factorization of 35 is simply 5 × 7. This is much simpler than 16, right? Now that we have the prime factorizations of both 16 (2^4) and 35 (5 × 7), we can use this information to find their least common multiple (LCM). Remember, the LCM is the smallest number that both 16 and 35 can divide into evenly. So, let's see how we can put these prime factors together to find that number.

Calculating the LCM of 16 and 35

Alright, we've got the prime factorizations of 16 (2^4) and 35 (5 × 7). Now it's time to put those factors to work and calculate the LCM. Remember, to find the LCM, we take the highest power of each prime factor that appears in either factorization and multiply them together. Looking at our prime factors, we have 2^4 from 16, and 5 and 7 from 35. So, we'll take 2^4, 5, and 7. Multiplying these together, we get 2^4 × 5 × 7 = 16 × 5 × 7. Let's do the math: 16 times 5 is 80, and 80 times 7 is 560. So, the LCM of 16 and 35 is 560. Awesome! We've completed the first part of our fraction LCM problem. Now, let's move on to the denominators and find their greatest common divisor (GCD). This is the next piece of the puzzle in figuring out the LCM of 16/63 and 35/48.

Step 2: Finding the GCD of the Denominators (63 and 48)

Now that we've conquered the numerators, let's turn our attention to the denominators: 63 and 48. This time, instead of finding the LCM, we need to find the greatest common divisor (GCD). The GCD is the largest number that can divide evenly into both 63 and 48. Just like with the LCM, we'll use prime factorization to help us out. First, we'll break down 63 and 48 into their prime factors. Then, we'll identify the common prime factors and use those to calculate the GCD. This step is just as crucial as finding the LCM of the numerators, so let's get to it and see how we can find the GCD of 63 and 48. Once we have this, we'll be one step closer to solving the entire problem!

Prime Factorization of 63

Let's start by finding the prime factorization of 63. We need to find the prime numbers that multiply together to give us 63. We can start by trying the smallest prime number, 2, but 63 isn't divisible by 2. Let’s try 3. 63 divided by 3 is 21. Now, we can divide 21 by 3 again, which gives us 7. And guess what? 7 is a prime number. So, the prime factorization of 63 is 3 × 3 × 7, or 3^2 × 7. See how breaking it down like this makes it easier? Now we know the prime building blocks of 63. Next up, we'll do the same for 48. Once we have both prime factorizations, we can compare them and find the greatest common divisor (GCD).

Prime Factorization of 48

Okay, let's tackle the prime factorization of 48. We're looking for the prime numbers that, when multiplied together, equal 48. Let's start with the smallest prime number, 2. 48 divided by 2 is 24. We can divide 24 by 2 again, which gives us 12. Divide 12 by 2, and we get 6. And finally, divide 6 by 2, and we get 3. So, we've broken 48 down to 2 × 2 × 2 × 2 × 3. That means 48 can be expressed as 2 to the power of 4 times 3, or 2^4 × 3. We’ve now got the prime factors of 48. We have the prime factorization for both 63 and 48, so it's time to compare them and find their greatest common divisor (GCD). This will help us wrap up the final step in finding the LCM of the original fractions.

Calculating the GCD of 63 and 48

Great! We've got the prime factorizations of 63 (3^2 × 7) and 48 (2^4 × 3). Now, let’s find the greatest common divisor (GCD). Remember, the GCD is the largest number that divides evenly into both 63 and 48. To find the GCD using prime factorizations, we look for the prime factors that both numbers have in common. Then, we take the lowest power of each common prime factor and multiply them together. Looking at our factorizations, we see that both 63 and 48 share the prime factor 3. The lowest power of 3 in these factorizations is 3^1 (since 63 has 3^2 and 48 has 3^1). So, the GCD of 63 and 48 is simply 3. Awesome! We've found the GCD of the denominators. Now we have all the pieces we need to calculate the LCM of the original fractions. Let's put it all together in the final step.

Step 3: Putting It All Together: The LCM of the Fractions

We've done the hard work, guys! We found the LCM of the numerators (16 and 35), which is 560, and the GCD of the denominators (63 and 48), which is 3. Now, to find the LCM of the fractions 16/63 and 35/48, we use the following formula:

LCM of fractions = (LCM of numerators) / (GCD of denominators)

Plugging in the values we found, we get:

LCM of 16/63 and 35/48 = 560 / 3

So, the LCM of the fractions 16/63 and 35/48 is 560/3. You can leave it like this, or if you prefer, you can convert it to a mixed number or a decimal. But as a fraction, 560/3 is our final answer. See? It wasn't so bad after all! We broke down the problem into smaller, manageable steps, and now we've got the solution. Let's recap what we did to make sure you’re totally confident with this process.

Recap: How to Find the LCM of Fractions

Okay, let's quickly recap the steps we took to find the LCM of the fractions 16/63 and 35/48. First, we identified the numerators (16 and 35) and the denominators (63 and 48). Then, we found the prime factorization of each numerator and denominator. This helped us calculate the LCM of the numerators, which turned out to be 560. Next, we found the prime factorization of each denominator and calculated the greatest common divisor (GCD), which was 3. Finally, we used the formula LCM of fractions = (LCM of numerators) / (GCD of denominators) to get our answer: 560/3. By breaking the problem down into these steps, we made it much easier to handle. Remember, the key to mastering these types of problems is practice. So, try out a few more examples, and you'll be a pro in no time! And that's it, guys! You've successfully learned how to find the LCM of fractions. Keep up the great work, and happy calculating!

Final Answer

So, to wrap it all up, the least common multiple (LCM) of the fractions 16/63 and 35/48 is 560/3. We got there by finding the LCM of the numerators (16 and 35) and the greatest common divisor (GCD) of the denominators (63 and 48), and then using the formula LCM of fractions = (LCM of numerators) / (GCD of denominators). Great job following along! Now you have a clear understanding of how to tackle these types of problems. Keep practicing, and you'll master it in no time!