Find A And B: (a, B) = 48, A*b = 6912

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Find a and b: (a, b) = 48, a*b = 6912

Hey guys! Today, we're diving into a fun math problem where we need to find two numbers, a and b, given their greatest common divisor (GCD) and their product. Specifically, we know that the GCD of a and b is 48, written as (a, b) = 48, and their product, a times b, is 6912. This is a classic number theory problem that combines the concepts of GCD and factorization. Let's break it down step-by-step so you can totally understand how to solve it.

Understanding the Problem

Before we jump into solving, let's make sure we all get what the problem is asking. The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest number that divides both a and b without leaving a remainder. In our case, 48 is the biggest number that divides both a and b. This means we can express a and b as multiples of 48. Also, we know that when we multiply a and b together, we get 6912. Our mission is to find the specific values of a and b that satisfy these conditions.

The key idea here is to express a and b in terms of their GCD. Since (a, b) = 48, we can write a = 48x and b = 48y, where x and y are coprime integers (meaning their GCD is 1). This representation is crucial because it simplifies the problem and allows us to work with smaller, relatively prime numbers. By using this representation and the given product a * b* = 6912, we can set up an equation in terms of x and y that will help us find their values, and consequently, the values of a and b.

Why is this representation so useful? Well, it leverages the fundamental property of the GCD. By expressing a and b as multiples of their GCD, we effectively strip away the common factors between them. This leaves us with x and y, which are relatively prime, meaning they share no common factors other than 1. This greatly simplifies the problem because we can now focus on finding these coprime integers, which are generally easier to work with than a and b themselves. This approach is a standard technique in number theory problems involving GCDs, and mastering it can help you tackle a wide range of similar problems.

Setting up the Equation

Alright, let's use what we know to set up an equation. We know that:

  • a = 48x
  • b = 48y
  • a * b = 6912

Now, substitute a and b in the third equation with their expressions in terms of x and y:

(48x) * (48y) = 6912

This simplifies to:

2304 * x * y = 6912

Now, divide both sides by 2304 to isolate x * y:

x * y = 6912 / 2304

x * y = 3

So, we've found that the product of x and y is 3. Remember, x and y must be coprime (their GCD is 1) because we factored out the greatest common divisor of a and b. This greatly narrows down the possibilities for x and y, making it easier to find their values.

The equation x * y = 3 is a crucial stepping stone. Since 3 is a prime number, its only factors are 1 and 3. This means that the possible pairs of integer values for x and y are quite limited. Given that x and y must be coprime (i.e., their greatest common divisor is 1), we can quickly identify the valid pairs. This simplification is a direct result of expressing a and b in terms of their GCD and then substituting those expressions into the product equation. It transforms a potentially complex problem into a manageable one with a clear path to the solution.

Finding Possible Values for x and y

Since x * y = 3 and x and y are coprime, the possible pairs for (x, y) are (1, 3) and (3, 1). We also need to consider the negative values, but since a and b are generally considered positive, we'll stick with the positive solutions for now. So we have two sets of solutions, which is very good and makes things simple!

Why do we only have two possible pairs? Because 3 is a prime number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. This means that the only way to express 3 as a product of two positive integers is 1 * 3. The condition that x and y must be coprime further restricts the possibilities. If x and y had a common factor greater than 1, they would not be coprime, and the condition (a, b) = 48 would not hold. Therefore, the only valid pairs are (1, 3) and (3, 1). This highlights the importance of understanding prime numbers and their properties when solving number theory problems.

Calculating a and b

Now that we have the possible values for x and y, we can find the corresponding values for a and b.

Case 1: (x, y) = (1, 3)

  • a = 48 * x = 48 * 1 = 48
  • b = 48 * y = 48 * 3 = 144

Case 2: (x, y) = (3, 1)

  • a = 48 * x = 48 * 3 = 144
  • b = 48 * y = 48 * 1 = 48

So, in both cases, we find that the numbers are 48 and 144. The order doesn't matter since the problem doesn't specify which number is a and which is b. This showcases how breaking down the problem into smaller parts, like finding x and y, makes the final calculation much easier.

Notice that the two cases essentially give us the same solution, just with a and b swapped. This is because the original problem is symmetric with respect to a and b; that is, interchanging a and b does not change the problem. Therefore, we can confidently conclude that the two numbers are 48 and 144, regardless of which one we call a and which one we call b. This symmetry is a useful observation that can sometimes simplify problem-solving in mathematics.

Verification

Let's verify our solution to make sure it satisfies the given conditions.

  1. GCD(48, 144): The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. The greatest common factor is indeed 48.
  2. Product: 48 * 144 = 6912

Both conditions are satisfied, so our solution is correct! Always remember to double-check your work to prevent errors.

Verifying the solution is a crucial step in any mathematical problem. It ensures that the answer obtained satisfies all the given conditions and constraints. In this case, we verified that the greatest common divisor of 48 and 144 is indeed 48, and their product is 6912, as required. This verification step confirms the correctness of our solution and provides confidence in the result. It also helps to catch any potential errors that may have occurred during the problem-solving process. Always make it a habit to verify your solutions whenever possible to ensure accuracy and completeness.

Conclusion

So, the numbers a and b are 48 and 144. We found this by using the properties of the greatest common divisor and setting up a simple equation. Hope you found this helpful! Keep practicing, and you'll become a pro at these types of problems. Understanding the relationships between numbers and their factors is super useful in all sorts of math, so keep at it!