Finding The Least Common Multiple (LCM) Of 105 And 135

by Jhon Lennon 55 views

Hey guys! Let's dive into the fascinating world of numbers and figure out how to find the Least Common Multiple (LCM) of 105 and 135. Finding the LCM is super useful in all sorts of real-life situations, like when you're trying to figure out when two events will happen at the same time or when you're working with fractions. The best part? We're going to use prime factorization, which is a cool way to break down numbers into their building blocks. It is not as scary as it sounds, I promise! So, grab your calculators (or just use your brainpower!), and let's get started. By the end of this, you'll be a pro at finding the LCM of any two numbers. Ready to learn something new?

Understanding Prime Factorization

Alright, before we jump into finding the LCM, let's make sure we're all on the same page about prime factorization. Think of prime factorization as the process of breaking down a number into a product of prime numbers. Remember those prime numbers? They're whole numbers greater than 1 that can only be divided by 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, and so on. For example, if you want to perform a prime factorization of the number 12, you must divide it by a prime number. 12 can be divided by 2, and the result is 6. 6 can be divided by 2, and the result is 3. 3 can be divided by 3, and the result is 1. Thus, the prime factorization of 12 is 2 x 2 x 3, or 2² x 3. Prime factorization is a fundamental concept in mathematics and is essential for understanding various mathematical operations, including finding the LCM and the Greatest Common Divisor (GCD). Mastering prime factorization will not only help you with LCM calculations but also give you a solid foundation for more advanced math concepts later on. So, understanding prime factorization is crucial to grasp the concepts behind LCM. Now, let's apply this concept to our numbers, 105 and 135.

Now, let's find the prime factorization of 105. We start by dividing 105 by the smallest prime number, which is 3. 105 divided by 3 is 35. 35 is not divisible by 3, so we try the next prime number, which is 5. 35 divided by 5 is 7. And finally, 7 is divisible by 7, which gives us 1. So, the prime factorization of 105 is 3 x 5 x 7. Easy peasy, right? Now, let's do the same for 135. Start by dividing 135 by 3. 135 divided by 3 is 45. 45 is also divisible by 3, so 45 divided by 3 is 15. Again, 15 is divisible by 3, and 15 divided by 3 is 5. Finally, 5 is divisible by 5, which gives us 1. The prime factorization of 135 is 3 x 3 x 3 x 5, or 3³ x 5. See, it's not as hard as it looks. The basic idea is just breaking down each number until all the factors are prime numbers. This skill comes in handy. It's like finding the secret ingredients that make up each number. Once you've found these secret ingredients (prime factors), you can find the LCM with ease.

Prime Factorization of 105 and 135

Now, let's find the prime factorization of 105 and 135 separately. This is a crucial step in our journey to find the Least Common Multiple (LCM). Remember, prime factorization is like breaking down a number into its prime building blocks. By doing this, we can easily identify the common and unique factors, which will help us calculate the LCM. So, let's get our hands dirty and break down these numbers! The prime factorization is the process of decomposing a composite number into a product of prime numbers. This is a fundamental concept in number theory and has various applications in mathematics. The prime factorization of a number is unique, which means that any composite number can be expressed as a product of prime numbers in only one way. Prime factorization helps to simplify complex calculations and find common divisors and multiples. Understanding prime factorization is essential for students and anyone working with numbers, as it is a crucial tool for solving various mathematical problems. It also lays the foundation for more advanced topics in mathematics, such as cryptography and computer science. So, let's delve deeper into prime factorization!

Prime Factorization of 105

To find the prime factorization of 105, start by dividing it by the smallest prime number that divides it evenly. In this case, it's 3. 105 divided by 3 equals 35. Now, consider 35. The smallest prime number that divides 35 evenly is 5. 35 divided by 5 equals 7. Finally, 7 is a prime number, so we divide it by itself. 7 divided by 7 equals 1. So, the prime factorization of 105 is 3 x 5 x 7. Nice! We've successfully broken down 105 into its prime factors.

Prime Factorization of 135

Now, let's do the same for 135. Start by dividing 135 by 3. 135 divided by 3 equals 45. 45 is also divisible by 3, so divide it by 3 again. 45 divided by 3 equals 15. Again, 15 is divisible by 3. 15 divided by 3 equals 5. Finally, 5 is a prime number, so divide it by 5. 5 divided by 5 equals 1. Therefore, the prime factorization of 135 is 3 x 3 x 3 x 5, which can also be written as 3³ x 5. Great job! We've also successfully found the prime factors of 135. The key is to keep dividing by prime numbers until you reach 1. This method ensures that you have identified all the prime factors of the number.

Finding the LCM Using Prime Factorization

Alright, now that we have the prime factorizations of both 105 (3 x 5 x 7) and 135 (3³ x 5), we can find the Least Common Multiple (LCM). To find the LCM using prime factorization, you need to identify all the prime factors from both numbers and take the highest power of each prime factor that appears in either factorization. Let me explain. The LCM is the smallest number that is a multiple of both 105 and 135. Think of it as the smallest number that both numbers can divide into evenly. So, here's how you do it step by step. First, list all the prime factors that appear in either factorization. In our case, the prime factors are 3, 5, and 7. Next, for each prime factor, identify the highest power it has in either factorization. For the prime factor 3, the highest power is 3³ (from the factorization of 135). For the prime factor 5, the highest power is 5¹ (it appears as 5 in both factorizations, which is the same as 5¹). And for the prime factor 7, the highest power is 7¹ (from the factorization of 105). Finally, multiply these highest powers together. So, the LCM of 105 and 135 is 3³ x 5¹ x 7¹ = 27 x 5 x 7 = 945. Boom! You've found the LCM. This method guarantees that the LCM will be divisible by both 105 and 135. The LCM is a fundamental concept in mathematics and has several practical applications. Understanding how to find the LCM is essential for students and anyone working with numbers, as it is a crucial tool for solving various mathematical problems. Let's practice with our numbers!

Step-by-Step Calculation

  1. Identify Prime Factors: List all the prime factors present in the prime factorizations of both 105 (3 x 5 x 7) and 135 (3³ x 5). The prime factors are 3, 5, and 7.
  2. Determine Highest Powers: For each prime factor, identify the highest power that appears in either factorization.
    • For 3: The highest power is 3³.
    • For 5: The highest power is 5¹.
    • For 7: The highest power is 7¹.
  3. Multiply the Highest Powers: Multiply these highest powers together to get the LCM.
    • LCM = 3³ x 5¹ x 7¹ = 27 x 5 x 7 = 945.

So, the LCM of 105 and 135 is 945. That means 945 is the smallest number that both 105 and 135 can divide into evenly. Knowing the LCM can be a lifesaver in real-world scenarios. For example, the LCM is super useful when working with fractions. If you want to add or subtract fractions with different denominators, you need to find the LCM of those denominators to find a common denominator. This makes it easier to perform the calculations. You will be a math whiz in no time!

Conclusion: The LCM of 105 and 135

And there you have it, folks! We've successfully found the Least Common Multiple (LCM) of 105 and 135 using prime factorization. We broke down the numbers into their prime factors, identified the highest powers of each factor, and multiplied them together to get our answer. Remember, the LCM is a fundamental concept in mathematics with tons of practical applications. From working with fractions to solving real-life problems, knowing how to find the LCM is a valuable skill. Keep practicing, and you'll become a pro in no time! So, the final answer is that the LCM of 105 and 135 is 945.

I hope this step-by-step guide has been helpful, and you've learned something new today. Remember, math can be fun and rewarding, especially when you understand the underlying concepts. Feel free to explore more problems and continue practicing to master the skill of finding the LCM. Keep up the amazing work, and keep exploring the wonderful world of numbers. Until next time, happy calculating, and keep those math muscles strong! We hope that this guide will help you in your math journey. Keep practicing and exploring the mathematical world!