Finding GCD And LCM With Ab=432 And [a,b]=3(a,b)

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Finding GCD and LCM with ab=432 and [a,b]=3(a,b)

Hey guys! Let's dive into this interesting math problem where we need to find the greatest common divisor (GCD) and the least common multiple (LCM) of two natural numbers, a and b. We're given two key pieces of information: the product of a and b is 432 (ab = 432), and the LCM of a and b is three times their GCD ([a,b] = 3(a,b)). Sounds like a fun puzzle, right? Let’s break it down step by step and solve it together!

Understanding the Basics: GCD and LCM

Before we jump into the solution, let’s quickly recap what GCD and LCM actually mean. The greatest common divisor (GCD), often denoted as (a,b), is the largest positive integer that divides both a and b without leaving a remainder. Think of it as the biggest number that can perfectly fit into both a and b. On the other hand, the least common multiple (LCM), denoted as [a,b], is the smallest positive integer that is divisible by both a and b. It's like the smallest number that both a and b can multiply into. Understanding these concepts is crucial because they are the building blocks for solving our problem.

GCD Explained

The GCD is super useful in many areas of mathematics, from simplifying fractions to solving Diophantine equations. To find the GCD, you can use methods like prime factorization or the Euclidean algorithm. Imagine you have two numbers, say 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors are 1, 2, 3, and 6. The greatest among these is 6, so the GCD(12, 18) = 6. This means 6 is the largest number that can divide both 12 and 18 evenly. Guys, remembering this basic concept helps in tackling more complex problems, and it’s essential for understanding the relationship between numbers.

LCM Explained

The LCM, on the other hand, helps us when we need to find a common multiple for two numbers. For instance, if you're adding fractions with different denominators, the LCM of the denominators gives you the smallest common denominator to work with. Let’s stick with our numbers 12 and 18. The multiples of 12 are 12, 24, 36, 48, 60, 72, and so on. The multiples of 18 are 18, 36, 54, 72, and so on. The common multiples are 36, 72, and so on. The smallest among these is 36, so the LCM(12, 18) = 36. This means 36 is the smallest number that both 12 and 18 can divide into without a remainder. Knowing how to find the LCM is invaluable in many mathematical scenarios, especially when dealing with fractions and ratios.

The Key Relationship: GCD, LCM, and the Product of Numbers

Here’s a fundamental theorem that will be our guiding light: For any two positive integers a and b, the product of their GCD and LCM is equal to the product of the numbers themselves. Mathematically, this is expressed as: (a,b) * [a,b] = a * b. This relationship is like a golden key that unlocks many problems involving GCD and LCM. It allows us to connect these two concepts with the actual numbers we're dealing with. In our case, we know that ab = 432, and [a,b] = 3(a,b). By using this theorem, we can set up equations that will help us find the values of (a,b) and [a,b]. Remember this formula, guys; it’s super handy!

How the Relationship Helps

This relationship simplifies our problem-solving process because it provides a direct link between GCD, LCM, and the product of the numbers. Instead of trying to find the GCD and LCM independently, we can use this equation to relate them. It’s like having a cheat code that makes the puzzle easier to solve. For example, if we know the product of two numbers and their GCD, we can easily find their LCM, and vice versa. This interrelation is what makes this theorem so powerful. In our specific problem, we are given that ab = 432 and [a,b] = 3(a,b). We can use the theorem to create an equation involving only (a,b), which we can then solve. This approach streamlines the solution and makes it more accessible.

Solving the Problem: Step-by-Step

Now, let's put our knowledge to work and solve the problem. We know that ab = 432 and [a,b] = 3(a,b). Our goal is to find both (a,b) and [a,b].

  1. Use the Key Relationship: We know that (a,b) * [a,b] = a * b. Substitute the given values: (a,b) * 3(a,b) = 432.
  2. Simplify the Equation: This simplifies to 3 * (a,b)^2 = 432.
  3. Solve for (a,b): Divide both sides by 3 to get (a,b)^2 = 144. Taking the square root of both sides gives us (a,b) = 12. Remember, we're looking for the positive root since GCD is a positive integer.
  4. Find [a,b]: Now that we know (a,b) = 12, we can use the given relationship [a,b] = 3(a,b). Substitute the value: [a,b] = 3 * 12 = 36.

So, we’ve found that the greatest common divisor (a,b) is 12, and the least common multiple [a,b] is 36. See, guys? It wasn’t so tough after all!

Detailed Steps

Let’s walk through each step with a little more detail to make sure we’ve got it all covered. First, we used the fundamental theorem that connects GCD, LCM, and the product of the numbers. This theorem is like the cornerstone of our solution. By substituting the given values into the equation (a,b) * [a,b] = a * b, we got (a,b) * 3(a,b) = 432. This is a crucial step because it sets up the equation we need to solve. Next, we simplified this equation. Multiplying (a,b) by 3(a,b) gives us 3 * (a,b)^2, so our equation becomes 3 * (a,b)^2 = 432. Now, we need to isolate (a,b)^2. To do this, we divide both sides of the equation by 3, which gives us (a,b)^2 = 144. The next step is to find the square root of both sides. The square root of 144 is 12, so we get (a,b) = 12. We only consider the positive root because GCD is always a positive integer. Finally, we use the relationship [a,b] = 3(a,b) to find the LCM. Substituting (a,b) = 12, we get [a,b] = 3 * 12 = 36. Thus, we’ve determined that the GCD is 12 and the LCM is 36.

Verifying the Solution

It's always a good idea to verify our solution to make sure everything checks out. We found that (a,b) = 12 and [a,b] = 36. Let’s check if these values satisfy our initial conditions.

  • Condition 1: ab = 432: We know that (a,b) * [a,b] = a * b. So, 12 * 36 should equal 432. Let’s multiply: 12 * 36 = 432. This condition is satisfied.
  • Condition 2: [a,b] = 3(a,b): We found [a,b] = 36 and (a,b) = 12. So, 3 * 12 should equal 36. Let’s multiply: 3 * 12 = 36. This condition is also satisfied.

Since both conditions are met, we can be confident that our solution is correct. Guys, always double-check your work! It helps catch any small errors and ensures you’re on the right track.

Why Verification is Important

Verifying the solution is a critical step in problem-solving, not just in mathematics but in any field. It’s like having a quality control check that ensures your answer is accurate and consistent with the given information. Without verification, you might end up with an incorrect solution, even if your steps seem logical. In our case, we verified that the GCD and LCM we found satisfy both the product condition (ab = 432) and the relationship between LCM and GCD ([a,b] = 3(a,b)). This process confirms that our values are correct and that our solution is valid. It also reinforces our understanding of the concepts involved. By verifying, we gain confidence in our answer and ensure we’re presenting accurate results. So, remember guys, verification is not just an extra step; it’s an essential part of the problem-solving process.

Finding Possible Values for a and b

Okay, so we know (a,b) = 12. This means both a and b are divisible by 12. We can write a = 12x and b = 12y, where x and y are coprime (i.e., their GCD is 1). Now, we also know that ab = 432. Substitute a and b: (12x)(12y) = 432. This simplifies to 144xy = 432. Divide both sides by 144 to get xy = 3.

Exploring Coprime Factors

Since x and y are coprime and their product is 3, the possible pairs for (x, y) are (1, 3) and (3, 1). Remember, coprime means that x and y have no common factors other than 1. This condition helps us narrow down the possible values for x and y. If we consider (x, y) = (1, 3), then a = 12 * 1 = 12 and b = 12 * 3 = 36. If we consider (x, y) = (3, 1), then a = 12 * 3 = 36 and b = 12 * 1 = 12. So, the possible pairs for (a, b) are (12, 36) and (36, 12). These pairs satisfy all the given conditions: their product is 432, their GCD is 12, and their LCM is 36. Exploring coprime factors is a valuable technique in number theory problems. It allows us to break down complex problems into simpler parts, making it easier to find the solutions.

Putting It All Together

Let’s recap the process. We started by expressing a and b in terms of their GCD, which is 12. We wrote a = 12x and b = 12y, where x and y are coprime. Then, we used the condition ab = 432 to find a relationship between x and y. Substituting our expressions for a and b into the equation, we got (12x)(12y) = 432. Simplifying this equation led us to 144xy = 432, and further simplification gave us xy = 3. Since x and y are coprime, we identified the possible pairs as (1, 3) and (3, 1). Finally, we substituted these pairs back into our expressions for a and b to find the possible values for (a, b). This step-by-step approach allowed us to systematically solve the problem and find all possible solutions. Guys, breaking down a problem into smaller steps makes it much more manageable and reduces the chance of errors.

Conclusion

So, guys, we've successfully found that (a,b) = 12 and [a,b] = 36. Additionally, we determined that the possible pairs for (a, b) are (12, 36) and (36, 12). This problem beautifully illustrates the relationship between GCD, LCM, and the product of two numbers. Remember the key theorem: (a,b) * [a,b] = a * b. It's a powerful tool in number theory!

Key Takeaways

To wrap up, let’s highlight the key takeaways from this problem. First, understanding the definitions of GCD and LCM is crucial. The GCD is the largest number that divides both given numbers, and the LCM is the smallest number that is divisible by both numbers. Second, the relationship (a,b) * [a,b] = a * b is a fundamental theorem that helps connect these concepts. This theorem allowed us to set up an equation and solve for the GCD and LCM. Third, always verify your solution to ensure it satisfies the given conditions. Verification helps catch errors and confirms the accuracy of your answer. Finally, breaking down the problem into smaller steps and using coprime factors can simplify the solution process. Guys, mastering these concepts and techniques will make you a pro at solving number theory problems! Keep practicing, and you'll become more confident in your mathematical abilities.