LCM And GCD Of 162, 58, And 90 A Step By Step Guide

Aug 3, 2025 by ADMIN 52 views

Unlocking the Mystery of LCM and GCD Finding the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of 162, 58, and 90

Hey guys! Ever found yourself scratching your head over finding the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of a set of numbers? It can seem daunting, but trust me, it's a super useful skill to have in your math arsenal. Today, we're going to break down how to find the LCM and GCD of 162, 58, and 90. So, buckle up, and let's dive in!

Understanding the Basics: LCM and GCD

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what LCM and GCD actually mean. The LCM, or Least Common Multiple, is the smallest positive integer that is perfectly divisible by each of the numbers in a set. Think of it as the smallest number that all your numbers can "fit into" evenly. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number that both 2 and 3 divide into without any remainder. On the other hand, the GCD, or Greatest Common Divisor (also sometimes called the Highest Common Factor, or HCF), is the largest positive integer that divides each of the numbers in a set without leaving a remainder. It's the biggest number that all your numbers can be divided by evenly. For instance, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 perfectly.

Knowing these definitions is crucial because LCM and GCD pop up in all sorts of math problems, from simplifying fractions to solving algebraic equations. Understanding how to find them efficiently can save you a ton of time and effort. We'll explore two popular methods for finding LCM and GCD: the prime factorization method and the Euclidean algorithm. Each has its strengths, and knowing both will give you a solid toolkit for tackling any LCM and GCD problem that comes your way. So, let’s get started and demystify these concepts together! We'll use the prime factorization method first, as it’s great for understanding the building blocks of numbers and how they relate to each other. Then, we can explore the Euclidean algorithm, which is particularly handy for larger numbers where prime factorization might get a bit cumbersome.

Method 1: Prime Factorization – Deconstructing Numbers

The prime factorization method is a powerful technique for finding both the LCM and GCD. The core idea behind this method is to break down each number into its prime factors. Prime factors are those numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, and so on). By expressing each number as a product of its prime factors, we can easily identify common factors and multiples, which are essential for finding the GCD and LCM. So, how do we actually do this? Let’s take a look at our numbers: 162, 58, and 90. We’ll start by finding the prime factors of each one.

For 162, we can start by dividing it by the smallest prime number, 2. 162 divided by 2 is 81. Now, 81 isn’t divisible by 2, so we move on to the next prime number, 3. 81 divided by 3 is 27. We can divide 27 by 3 again to get 9, and then divide 9 by 3 to get 3. Finally, 3 is a prime number, so we stop there. This means the prime factorization of 162 is 2 × 3 × 3 × 3 × 3, or 2 × 3⁴. Next, let’s tackle 58. We can divide 58 by 2 to get 29. 29 is a prime number, so we’re done. The prime factorization of 58 is simply 2 × 29. Lastly, let’s break down 90. We can divide 90 by 2 to get 45. 45 isn’t divisible by 2, so we try 3. 45 divided by 3 is 15. We can divide 15 by 3 again to get 5. 5 is a prime number, so we stop. The prime factorization of 90 is 2 × 3 × 3 × 5, or 2 × 3² × 5. Now that we have the prime factorizations, we can use them to find both the GCD and LCM.

The beauty of prime factorization lies in its systematic approach. By breaking numbers down to their most basic components, we reveal the underlying structure that governs their divisibility. This is super helpful when dealing with larger numbers or sets of numbers where finding common factors might not be immediately obvious. The prime factorization method not only helps in finding GCD and LCM but also provides a deeper understanding of number theory concepts. It’s a foundational tool that will serve you well in various mathematical contexts. Remember, practice makes perfect, so try breaking down different numbers into their prime factors to get comfortable with the process. Once you master this method, you’ll find that many number-related problems become much easier to solve. Plus, it’s kind of like detective work, uncovering the hidden building blocks of numbers! So, keep practicing and enjoy the process of unraveling the mysteries of numbers.

Finding the GCD Using Prime Factors

Now that we have the prime factorizations of 162, 58, and 90, let's use them to find the GCD. Remember, the GCD is the largest number that divides all the given numbers without leaving a remainder. To find the GCD using prime factors, we look for the prime factors that are common to all the numbers and then multiply them together, taking the lowest power of each common prime factor. Let's list the prime factorizations again for clarity:

162 = 2 × 3⁴
58 = 2 × 29
90 = 2 × 3² × 5

Looking at these, we can see that the only prime factor common to all three numbers is 2. The lowest power of 2 that appears in these factorizations is 2¹, which is simply 2. There are no other common prime factors among 162, 58, and 90. Therefore, the GCD of 162, 58, and 90 is 2. Isn't that neat? By breaking down the numbers into their primes, we made it super easy to spot the greatest common divisor. This method really shines when dealing with larger numbers where common factors might not be immediately apparent. You can see how the prime factorization acts like a magnifying glass, highlighting the shared components of the numbers. It's like understanding the DNA of each number, which makes it straightforward to identify common ancestry.

Understanding this process not only helps you find the GCD but also reinforces your understanding of what the GCD actually represents. It's not just a number; it's a fundamental relationship between the numbers you're working with. By knowing the GCD, you can simplify fractions, solve equations, and tackle other mathematical challenges with greater confidence. So, the next time you need to find the GCD, remember the power of prime factorization. It's a reliable and insightful method that will help you unlock the common ground between numbers. Keep practicing, and you'll become a GCD-finding pro in no time!

Calculating the LCM Using Prime Factors

Alright, guys, let's switch gears and figure out the LCM of 162, 58, and 90 using those same prime factorizations we found earlier. Remember, the LCM is the smallest number that all the given numbers divide into evenly. To find the LCM using prime factors, we need to identify all the prime factors that appear in any of the numbers and then multiply them together, this time taking the highest power of each prime factor. Let's bring back our prime factorizations for reference:

162 = 2 × 3⁴
58 = 2 × 29
90 = 2 × 3² × 5

Now, let's go through each prime factor. We have 2, 3, 5, and 29. The highest power of 2 that appears is 2¹ (which is just 2). The highest power of 3 is 3⁴ (from the factorization of 162). We also have 5¹ (which is 5) from the factorization of 90, and 29¹ (which is 29) from the factorization of 58. To find the LCM, we multiply these highest powers together: LCM = 2 × 3⁴ × 5 × 29. Let's do the math: 3⁴ is 81, so we have LCM = 2 × 81 × 5 × 29. Multiplying 2 by 81 gives us 162. Then, 162 times 5 is 810. Finally, 810 times 29 is 23490. So, the LCM of 162, 58, and 90 is 23490. That's a pretty big number, but it makes sense because it needs to be divisible by all three original numbers.

This method of finding the LCM by using prime factors is not only accurate but also gives us a clear picture of why the LCM is what it is. Each prime factor is included to the highest power necessary to ensure that the LCM is divisible by each of the original numbers. Think of it like building a number from its fundamental components, making sure you have enough of each component to accommodate all the numbers in your set. The LCM is like the ultimate common ground, the smallest number that encompasses all the others. This understanding is super useful in various mathematical contexts, such as adding fractions with different denominators or solving problems involving periodic events. By mastering this technique, you'll not only be able to calculate the LCM efficiently but also grasp the underlying principles that make it work. So, keep practicing and see how this powerful tool can simplify your math journey!

Method 2: Euclidean Algorithm – A Speedy Approach for GCD

Okay, now let's explore another fantastic method for finding the GCD: the Euclidean algorithm. This method is particularly useful when you're dealing with large numbers, as it avoids the need for prime factorization. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. We repeat this process until one of the numbers becomes zero, and the other number is then the GCD. Sounds a bit abstract? Let's walk through it step by step with our numbers, 162, 58, and 90.

First, we'll find the GCD of 162 and 58. We divide 162 by 58 and find the remainder. 162 divided by 58 is 2 with a remainder of 46. So, we replace 162 with 58 and 58 with 46. Now we have 58 and 46. We repeat the process: 58 divided by 46 is 1 with a remainder of 12. So, we replace 58 with 46 and 46 with 12. Now we have 46 and 12. 46 divided by 12 is 3 with a remainder of 10. We replace 46 with 12 and 12 with 10. Now we have 12 and 10. 12 divided by 10 is 1 with a remainder of 2. We replace 12 with 10 and 10 with 2. Now we have 10 and 2. 10 divided by 2 is 5 with a remainder of 0. Bingo! We've reached a remainder of 0, so the GCD of 162 and 58 is 2. Now that we have the GCD of 162 and 58, which is 2, we need to find the GCD of this result and the remaining number, 90. So, we'll find the GCD of 2 and 90. This is pretty straightforward: since 2 divides 90 evenly, the GCD of 2 and 90 is 2. Therefore, the GCD of 162, 58, and 90 is 2.

The Euclidean algorithm might seem a bit like a dance of numbers, but it's incredibly efficient and elegant. It avoids the sometimes tedious process of prime factorization, especially when dealing with large numbers that have complex prime factors. The beauty of this algorithm lies in its simplicity and its guaranteed convergence. You keep reducing the numbers until you reach a point where the GCD reveals itself. It's like peeling away layers to find the common core. This method is a testament to the power of mathematical thinking, showing how a simple idea can lead to a powerful solution. Mastering the Euclidean algorithm is a valuable addition to your math toolkit. It's not only useful for finding the GCD but also for understanding other concepts in number theory. So, practice this method with different sets of numbers, and you'll become a GCD-finding wizard in no time!

Wrapping Up: Putting It All Together

So, guys, we've journeyed through finding the LCM and GCD of 162, 58, and 90 using two powerful methods: prime factorization and the Euclidean algorithm. We saw how prime factorization helps us break down numbers into their fundamental building blocks, making it easier to identify common factors and multiples. This method is great for understanding the underlying structure of numbers and is particularly useful when the prime factors are relatively small and easy to find. On the other hand, we explored the Euclidean algorithm, a speedy and elegant approach for finding the GCD, especially when dealing with larger numbers where prime factorization might be cumbersome. This method relies on a simple yet profound principle: repeatedly replacing the larger number with the remainder of the division until we reach a GCD.

We found that the GCD of 162, 58, and 90 is 2, meaning 2 is the largest number that divides all three numbers evenly. We also calculated the LCM to be 23490, which is the smallest number that all three numbers divide into without any remainder. These values give us a deeper understanding of the relationships between these numbers and their divisibility properties. Knowing how to find the LCM and GCD is super valuable in various mathematical contexts, from simplifying fractions to solving complex equations. These skills are not just about crunching numbers; they're about developing a strong foundation in number theory and mathematical reasoning.

As you continue your mathematical journey, remember that practice is key. The more you work with these methods, the more comfortable and confident you'll become. Try finding the LCM and GCD of different sets of numbers, and challenge yourself with more complex problems. Don't be afraid to explore different approaches and find what works best for you. Math is like a puzzle, and each problem is a chance to sharpen your skills and unlock new insights. So, keep exploring, keep practicing, and most importantly, keep enjoying the process of learning! You've got this!

Final Answer

The final answer is GCD = 2 and LCM = 23490