GCD(24, 72) / LCM(3, 4): A Step-by-Step Solution

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Hey guys! Let's dive into a super cool math problem today: figuring out the value of GCD(24, 72) divided by LCM(3, 4). This might sound a bit intimidating at first, but trust me, we'll break it down into easy-peasy steps. So, grab your thinking caps, and let’s get started!

What are GCD and LCM?

Before we jump into solving the problem, it's essential to understand what GCD and LCM actually mean. These are fundamental concepts in number theory, and they're super useful in various mathematical contexts.

GCD: The Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder. Think of it as the biggest number that can fit perfectly into all the numbers you're considering. For example, if we're looking at the numbers 12 and 18, the GCD is 6 because 6 is the largest number that divides both 12 and 18 evenly.

Finding the GCD is crucial in many areas, such as simplifying fractions and solving Diophantine equations. There are several methods to calculate the GCD, including listing factors, using prime factorization, and the Euclidean algorithm. We'll touch on these methods later when we tackle our problem.

LCM: The Least Common Multiple

On the flip side, we have the Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of two or more numbers. In simpler terms, it's the smallest number that each of your original numbers can divide into without any remainder. For instance, if we consider the numbers 4 and 6, the LCM is 12 because 12 is the smallest number that both 4 and 6 divide into evenly.

The LCM is particularly useful when you're dealing with fractions with different denominators, as it helps you find a common denominator to perform operations like addition and subtraction. Like the GCD, the LCM can be found using various methods, such as listing multiples, using prime factorization, and leveraging the relationship between GCD and LCM.

Step 1: Calculate the GCD(24, 72)

Okay, now that we've got a solid grasp of GCD and LCM, let's roll up our sleeves and calculate the GCD of 24 and 72. There are a couple of ways we can do this, but let's start with the prime factorization method. It's a classic and reliable way to find the GCD.

Prime Factorization Method

First, we need to break down both 24 and 72 into their prime factors. Prime factors are prime numbers that, when multiplied together, give you the original number. Remember, a prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).

Let's break it down:

  • 24 = 2 × 2 × 2 × 3 = 2³ × 3
  • 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Now that we have the prime factorizations, we can find the GCD by identifying the common prime factors and their lowest powers. In this case, both 24 and 72 share the prime factors 2 and 3.

The lowest power of 2 present in both factorizations is 2³. The lowest power of 3 present in both factorizations is 3¹.

So, the GCD(24, 72) is the product of these lowest powers:

GCD(24, 72) = 2³ × 3¹ = 8 × 3 = 24

Therefore, the GCD(24, 72) is 24. This means that 24 is the largest number that divides both 24 and 72 without leaving a remainder. Cool, right?

Step 2: Calculate the LCM(3, 4)

Alright, we've nailed the GCD part. Now, let's shift our focus to finding the LCM of 3 and 4. Just like with the GCD, we have a few methods at our disposal. Let's stick with the prime factorization method for consistency.

Prime Factorization Method (Again!)

We start by breaking down 3 and 4 into their prime factors:

  • 3 = 3 (3 is already a prime number)
  • 4 = 2 × 2 = 2²

To find the LCM, we identify all the unique prime factors and their highest powers present in either factorization. In this case, we have the prime factors 2 and 3.

The highest power of 2 is 2². The highest power of 3 is 3¹.

So, the LCM(3, 4) is the product of these highest powers:

LCM(3, 4) = 2² × 3¹ = 4 × 3 = 12

Therefore, the LCM(3, 4) is 12. This means that 12 is the smallest number that both 3 and 4 divide into evenly. Awesome!

Step 3: Divide the GCD by the LCM

We're in the home stretch now! We've calculated the GCD(24, 72) and the LCM(3, 4). All that's left to do is divide the GCD by the LCM. Easy peasy, right?

We found that:

  • GCD(24, 72) = 24
  • LCM(3, 4) = 12

Now, we just perform the division:

Result = GCD(24, 72) / LCM(3, 4) = 24 / 12 = 2

So, the final answer is 2! We've successfully calculated the value of GCD(24, 72) / LCM(3, 4).

Alternative Methods for GCD and LCM

While we primarily used the prime factorization method, it's worth knowing there are other ways to find the GCD and LCM. These methods can be particularly useful in different situations, and understanding them will make you a math whiz!

Euclidean Algorithm for GCD

The Euclidean algorithm is a super-efficient method for finding the GCD of two numbers. It involves repeatedly applying the division algorithm until you reach a remainder of 0. The GCD is the last non-zero remainder.

Let's use the Euclidean algorithm to find GCD(24, 72):

  1. Divide 72 by 24: 72 = 24 × 3 + 0

Since the remainder is 0, the GCD is the last divisor, which is 24. See how quick that was?

Listing Multiples for LCM

Another way to find the LCM is by listing the multiples of each number until you find a common multiple. This method is straightforward, especially for smaller numbers.

Let's find the LCM(3, 4) using this method:

  • Multiples of 3: 3, 6, 9, 12, 15, ...
  • Multiples of 4: 4, 8, 12, 16, ...

The smallest common multiple is 12, which confirms our earlier result.

Why are GCD and LCM Important?

You might be wondering, "Okay, this is cool, but why should I care about GCD and LCM?" Well, these concepts are more than just mathematical curiosities. They have practical applications in various real-world scenarios and are fundamental in many areas of mathematics.

Real-World Applications

  • Simplifying Fractions: The GCD is used to simplify fractions to their lowest terms. For example, if you have the fraction 24/72, you can divide both the numerator and the denominator by their GCD (which is 24) to get the simplified fraction 1/3.
  • Scheduling Problems: The LCM can be used to solve scheduling problems where you need to find when events will coincide. For instance, if one event happens every 3 days and another happens every 4 days, the LCM (12) tells you that they will coincide every 12 days.
  • Computer Science: GCD and LCM are used in cryptography, data compression, and other areas of computer science.

Mathematical Significance

  • Number Theory: GCD and LCM are fundamental concepts in number theory and are used to prove many theorems and results.
  • Algebra: They appear in algebraic manipulations and are used to solve equations and simplify expressions.

Conclusion

So, there you have it! We've successfully calculated GCD(24, 72) / LCM(3, 4) and found the answer to be 2. We've also explored what GCD and LCM mean, how to calculate them using different methods, and why they're important in both real-world applications and theoretical mathematics.

I hope this comprehensive guide has helped you understand these concepts better. Remember, math can be fun when you break it down step by step. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to dive deeper into this topic, feel free to ask. Happy calculating, guys!