Finding Natural Numbers A, B, C With Maximum B Value

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Finding Natural Numbers a, b, c with Maximum b Value

Hey guys! Today, we're diving into a super interesting math problem where we need to figure out some natural numbers. Natural numbers are just your regular counting numbers (1, 2, 3, and so on). The challenge here is to find three of these numbers – let's call them a, b, and c – that fit certain rules. Specifically, we know that when we multiply a and b, we should get 168. And when we multiply b and c, we should end up with 350. But here's the kicker: we want to find the biggest possible number for b. This involves a bit of number magic, and by the end of this guide, you’ll not only understand how to solve this specific problem but also the general principles behind it, which you can apply to all sorts of similar puzzles. So, let's get our thinking caps on and jump into the fascinating world of number theory!

Understanding the Problem: Natural Numbers and Maximum Value

The heart of this problem lies in figuring out the natural numbers a, b, and c, with a special focus on maximizing the value of b. Let's break down what this means and why it's important.

Firstly, natural numbers are the positive integers we use for counting (1, 2, 3, ...). They don't include zero, fractions, or negative numbers. Understanding this basic definition is crucial because it limits the possible solutions we can consider. We're not dealing with decimals or fractions here, only whole numbers.

Now, let's consider the two equations we have: a * b = 168 and b * c = 350. These equations tell us that b is a factor (or divisor) of both 168 and 350. A factor is a number that divides evenly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 perfectly.

The challenge here isn't just to find any common factor for 168 and 350, but to find the greatest common factor (GCF). Why? Because we want the largest possible value for b. If b is larger, it means a and c will be smaller, still fitting our equations, but maximizing b's value. Imagine you have a fixed area for a rectangle (like our products 168 and 350). If you want one side (b) to be as long as possible, you need to adjust the other side accordingly.

So, in essence, we're on a quest to discover the biggest number that divides both 168 and 350. This number will be our b, and from there, we can easily find a and c. Grasping this concept is the first big step in solving the problem, and it sets the stage for the next part: figuring out how to actually find this GCF. Let's get to it!

Finding the Greatest Common Factor (GCF)

Okay, so we know that to solve our problem, we need to find the greatest common factor (GCF) of 168 and 350. But how exactly do we do that? There are a couple of cool methods we can use, and I'm going to walk you through both so you can choose the one that clicks best with you. Let's make finding GCFs super easy!

Method 1: Listing Factors

One way to find the GCF is by listing all the factors of each number and then identifying the largest one they have in common. It’s a bit like comparing two piles of puzzle pieces to see which is the biggest piece that fits in both puzzles.

Let's start with 168. We need to think of all the numbers that divide 168 without leaving a remainder. These are:

1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.

That's quite a list, but don't worry, we’re just being thorough! Now, let's do the same for 350. The factors of 350 are:

1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, and 350.

Now comes the fun part – comparing the two lists. Look for the numbers that appear in both lists. You'll see that 1, 2, 7, and 14 are common factors. But we’re not just looking for any common factor; we want the greatest one. So, among these, 14 is the largest number.

Therefore, the GCF of 168 and 350 is 14. This method is pretty straightforward and helps you visualize what factors are, but it can be a bit time-consuming, especially with larger numbers. That's where our second method comes in handy. Let’s check it out!

Method 2: Prime Factorization

The prime factorization method is another fantastic way to find the GCF, especially when dealing with bigger numbers. Think of it as breaking down each number into its most basic building blocks and then seeing what they have in common.

Prime numbers are numbers that have only two factors: 1 and themselves (like 2, 3, 5, 7, 11, and so on). Prime factorization is the process of expressing a number as a product of its prime factors. So, let's start by finding the prime factorization of 168.

  • We can start by dividing 168 by the smallest prime number, 2: 168 ÷ 2 = 84
  • Then, divide 84 by 2 again: 84 ÷ 2 = 42
  • Keep going: 42 ÷ 2 = 21
  • Now, 21 isn't divisible by 2, so we move to the next prime number, 3: 21 ÷ 3 = 7
  • 7 is a prime number itself, so we stop here.

So, the prime factorization of 168 is 2 × 2 × 2 × 3 × 7, or 2³ × 3 × 7.

Now, let's do the same for 350:

  • 350 ÷ 2 = 175
  • 175 isn't divisible by 2 or 3, so we try 5: 175 ÷ 5 = 35
  • Divide by 5 again: 35 ÷ 5 = 7
  • And again, 7 is prime, so we're done.

The prime factorization of 350 is 2 × 5 × 5 × 7, or 2 × 5² × 7.

Now comes the cool part: to find the GCF, we identify the common prime factors and their lowest powers. Both numbers have 2 and 7 as prime factors. The lowest power of 2 in both factorizations is 2¹ (just 2), and they both have 7¹ (just 7).

So, the GCF is 2 × 7 = 14. Ta-da! We arrived at the same answer as before, but with a slightly different approach. This method is super useful when you're dealing with larger numbers because it breaks the problem down into smaller, more manageable parts.

With both methods in your toolkit, you can now confidently find the GCF of any two numbers. And remember, finding the GCF is the key to unlocking the solution to our original problem. So, let's use this knowledge to find those natural numbers a, b, and c!

Solving for a, b, and c

Alright, we've done the groundwork! We know that the greatest common factor (GCF) of 168 and 350 is 14. Remember, this GCF is the largest possible value for b in our problem. So, we've already found one of our numbers: b = 14! Feels good to have made some progress, right?

Now that we know b, finding a and c is going to be a piece of cake. We just need to use the two equations we were given at the start: a * b = 168 and b * c = 350. Since we know b, we can plug it into these equations and solve for a and c.

Finding a

Let's start with the equation a * b = 168. We know b = 14, so we can rewrite the equation as a * 14 = 168. To find a, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 14:

a = 168 / 14

Now, we just need to do the division. If you're comfortable with long division, go for it! If not, a calculator works perfectly too. 168 divided by 14 is 12. So, we've found another number: a = 12! See how easy that was once we knew b?

Finding c

Next up, let's find c. We'll use the second equation, b * c = 350. Again, we know b = 14, so we can rewrite the equation as 14 * c = 350. Just like before, we need to isolate c. We do this by dividing both sides of the equation by 14:

c = 350 / 14

Now, let's do the division. 350 divided by 14 is 25. So, we've found our last number: c = 25! Awesome!

The Solution

We did it! We've successfully found the values of a, b, and c that fit our problem. Let's recap:

  • a = 12
  • b = 14
  • c = 25

These natural numbers satisfy the conditions a * b = 168, b * c = 350, and b has the largest possible value. We solved this by understanding the importance of the greatest common factor and then using it to find the individual numbers.

But hold on, we're not quite done yet. It’s always a good idea to double-check our work to make sure everything adds up correctly. Let's do that now!

Verifying the Solution

Okay, so we've found our numbers: a = 12, b = 14, and c = 25. But before we pat ourselves on the back completely, let's make absolutely sure our solution is correct. Think of it as the final polish on a masterpiece – it’s that important! To verify our solution, we need to plug these values back into our original equations and see if they hold true. This is a crucial step in problem-solving because it catches any little mistakes we might have made along the way.

Checking a * b = 168

Our first equation is a * b = 168. Let's substitute our values for a and b:

12 * 14 = ?

If you multiply 12 by 14, you get 168. So, the left side of the equation equals the right side. That's a good start! It means our values for a and b work together correctly in the first equation.

Checking b * c = 350

Now, let's check our second equation: b * c = 350. Again, we'll substitute our values, this time for b and c:

14 * 25 = ?

If you multiply 14 by 25, you get 350. Awesome! This means our values for b and c also work together correctly in the second equation.

Final Verification

We've checked both equations, and our values for a, b, and c make both of them true. This gives us a high level of confidence that our solution is correct. But there's one more thing we should think about: Did we really find the largest possible value for b?

Remember, we found b by calculating the greatest common factor of 168 and 350. By definition, the GCF is the largest number that divides both 168 and 350. So, we can be sure that there's no larger value for b that would work in both equations with natural numbers for a and c.

Conclusion of Verification

So, after carefully checking our work, we can confidently say that our solution is correct. We found the natural numbers a = 12, b = 14, and c = 25, which satisfy the given conditions, and we maximized the value of b. Great job, guys! This meticulous approach to problem-solving – finding a solution and then verifying it – is a valuable skill in math and beyond.

Conclusion

Alright guys, we made it! We tackled a pretty cool math problem today, figuring out how to find natural numbers a, b, and c that fit specific equations, while also making sure that b was as big as possible. This wasn't just about getting to the right answer; it was about understanding the process of problem-solving. We used some key concepts, like natural numbers, factors, and the greatest common factor (GCF), to break down the problem into manageable steps.

We started by understanding the problem and identifying that we needed to find the GCF of 168 and 350 to maximize b. Then, we explored two different methods for finding the GCF: listing factors and prime factorization. Both methods are super useful, and knowing both gives you flexibility in how you approach similar problems in the future.

Once we found that the GCF (and therefore the maximum value for b) was 14, we used this information to easily calculate a and c using our original equations. And finally, we didn't just stop there – we verified our solution by plugging the values back into the equations to make sure everything checked out. This step is so important because it ensures accuracy and builds confidence in your solution.

This kind of problem-solving approach is like learning a superpower. It's not just about memorizing formulas or procedures; it's about developing a way of thinking that you can apply to all sorts of challenges. Whether you're dealing with math problems, coding challenges, or real-life situations, the ability to break things down, use the right tools, and verify your results is invaluable.

So, give yourselves a big pat on the back for working through this with me. You've not only learned how to solve this particular problem, but you've also strengthened your problem-solving skills. Keep practicing, keep exploring, and most importantly, keep having fun with math! Who knows what other number mysteries you'll be able to unlock?