Solving Math Problems: [a,b]=144 And A•b=6912

Hey math enthusiasts! Today, we're diving into a fun problem that's perfect for 6th graders. We're going to solve the equation where the least common multiple (LCM) of two numbers, a and b, is 144, and their product (a times b) is 6912. Sounds tricky? Don't worry, we'll break it down step-by-step to make it super easy to understand. We're not just aiming to find the solution; we're also going to explore some cool mathematical concepts along the way. Get ready to flex those brain muscles and have a blast with numbers! Let's get started, guys!

Understanding the Basics: LCM and Product

First things first, let's make sure we're all on the same page. The Least Common Multiple (LCM) of two numbers is the smallest number that both numbers can divide into evenly. Think of it like this: if you have two trains, the LCM is the point where they'll both arrive at the station at the same time. The product of two numbers is simply the result you get when you multiply them together. It's like adding the same number over and over, but much faster. In our problem, we know the LCM of a and b is 144, and their product (a * b*) is 6912. This information is like the secret ingredients to our mathematical recipe. Now, how do we use these ingredients to find the values of a and b? Well, let’s dig in deeper and unravel the puzzle. To make things clearer, let's look at a simpler example. Imagine the LCM of two numbers is 6 and their product is 12. You could quickly guess that the numbers are 2 and 6 or 3 and 4, since they both satisfy the conditions. Our problem is slightly more complex, but the underlying principle remains the same. Knowing the LCM and the product gives us valuable insights into the relationship between the two numbers. The main goal here is to find the two numbers. This is one of the most important aspects when you are doing math. It helps you keep track of all the steps that you are taking to find the final result. In short, always remember the goal.

The Relationship Between LCM and Product

There's a cool relationship between the LCM and the product of two numbers. This relationship is a fundamental concept in number theory. For any two numbers, the product of the numbers is always equal to the product of their LCM and their Greatest Common Divisor (GCD). This can be expressed as: a * b* = LCM(a, b) * GCD(a, b). In our case, we know a * b* = 6912 and LCM(a, b) = 144. So, we can find the GCD of a and b by using the formula. By understanding this relationship, we can determine the factors that make up each number. We're going to use this relationship to get closer to solving our problem. The GCD, or Greatest Common Divisor, is the largest number that divides both a and b without leaving a remainder. It's like finding the biggest common block we can use to build both numbers. This formula is super handy because it connects the LCM, GCD, and product in a neat little package. This relationship provides a clear pathway to solve the equation. This is the main core and should be remembered.

Finding the Greatest Common Divisor (GCD)

Now, let's calculate the Greatest Common Divisor (GCD) of a and b. We know that a * b* = 6912 and LCM(a, b) = 144. Using the formula a * b* = LCM(a, b) * GCD(a, b), we can rearrange it to find the GCD: GCD(a, b) = (a * b*) / LCM(a, b). Plugging in our values: GCD(a, b) = 6912 / 144. When we do the division, we get GCD(a, b) = 48. This means the largest number that divides both a and b is 48. This is a crucial piece of information! The GCD helps us understand the common factors that a and b share. It's like finding the common building blocks of our numbers. Remember, this step is all about finding the common ground between our two numbers. This will help you find the values for a and b. This understanding is the key to unlock the problem.

Using the GCD to Find the Numbers

Since the GCD of a and b is 48, we can write a = 48x and b = 48y, where x and y are co-prime numbers (meaning they have no common factors other than 1). Because a and b are both divisible by 48, we've essentially factored out the common element. Now, let's substitute these values into the product equation: a * b* = 6912 becomes (48x) * (48y) = 6912. Simplifying, we get 2304 x * y* = 6912. Dividing both sides by 2304 gives us x * y* = 3. This means that the product of x and y is 3. Since x and y are co-prime, the only possible integer values for x and y are 1 and 3 (or 3 and 1). So, either x = 1 and y = 3, or x = 3 and y = 1. This tells us what values we should use to find the final result.

Calculating the Values of a and b

Now, let's calculate the values of a and b using the values we've found for x and y. If x = 1 and y = 3, then a = 48 * 1 = 48 and b = 48 * 3 = 144. Or, if x = 3 and y = 1, then a = 48 * 3 = 144 and b = 48 * 1 = 48. In both cases, the numbers are 48 and 144. Let's check if these numbers satisfy our original conditions. The LCM of 48 and 144 is indeed 144, and their product is 48 * 144 = 6912, which matches the given information. Thus, we have successfully found the values of a and b. Congratulations, you've solved the problem, guys!

Checking the Solution

It’s always a good practice to verify your answers. Let’s double-check our solution. We found that a and b are 48 and 144. First, let's calculate the LCM of 48 and 144. The multiples of 48 are 48, 96, 144, 192... and the multiples of 144 are 144, 288... The smallest common multiple is 144, which matches the given LCM. Next, let’s find the product of 48 and 144: 48 * 144 = 6912. This also matches the given product. Our calculations are correct, and we can confidently say that the solution to the problem is a = 48 and b = 144 (or vice-versa). This step ensures the reliability of our solution. This is how you confirm that the result is correct.

Conclusion: You Did It!

Congratulations! You've successfully solved the math problem. You've learned about the relationship between LCM, GCD, and the product of two numbers, and you've applied these concepts to find the values of a and b. Solving this problem required several steps, but by breaking it down and understanding each part, we made it manageable. Remember, math is like a puzzle. With each problem, you're learning new tools and strategies to solve more complex puzzles. Keep practicing, keep exploring, and most importantly, keep having fun with numbers! You can do a lot more math problems by learning this. Keep in mind that math can be fun! This is something that you should always remember. With practice, you’ll become a math pro in no time! Keep practicing!