Distinct Sets Of Natural Numbers Problem Solving A × B × C = 2310

Hey guys! Ever stumbled upon a math problem that seems like a puzzle? Today, we're diving into a cool problem that involves finding distinct sets of natural numbers. It’s one of those questions that might pop up in math discussions or even on a test, so let’s break it down together. Our mission is to figure out how many different sets {a, b, c} of three natural numbers satisfy the condition a × b × c = 2310, and here's the kicker – a, b, and c must all be different from each other.

Understanding the Problem

So, first things first, let's make sure we're all on the same page. We're looking for natural numbers, which are the positive whole numbers (1, 2, 3, and so on). We need to find three of these numbers (a, b, and c) that, when multiplied together, give us 2310. But, there's a twist! The order doesn't matter (that's what the set notation {a, b, c} tells us), and the numbers must be distinct (different from each other).

Prime Factorization: The Key to Unlocking the Problem

The secret sauce to solving this kind of problem is prime factorization. Prime factorization is like breaking down a number into its prime building blocks. Prime numbers are numbers that are only divisible by 1 and themselves (like 2, 3, 5, 7, etc.).

So, let's factorize 2310. We can start by dividing it by the smallest prime number, 2:

2310 ÷ 2 = 1155

Now, 1155 isn't divisible by 2, so let's try the next prime number, 3:

1155 ÷ 3 = 385

Great! Now we have 385. Let's keep going. It's not divisible by 3, so let's try 5:

385 ÷ 5 = 77

We're getting there! 77 isn't divisible by 5, so let's try 7:

77 ÷ 7 = 11

And finally, 11 is a prime number itself. So, we've broken down 2310 into its prime factors:

2310 = 2 × 3 × 5 × 7 × 11

This is awesome! We now know that 2310 is the product of these five prime numbers. This is a crucial step in figuring out our sets.

Forming the Sets: The Combinatorial Challenge

Now comes the fun part: figuring out how to combine these prime factors to form our sets {a, b, c}. Remember, we need three numbers, and they all have to be different. Each of these numbers (a, b, and c) will be a product of one or more of these prime factors. This is where it starts to feel like a combinatorial puzzle, and that's super exciting.

To solve this, we need to think about how we can group these prime factors. Since we have five prime numbers and we need three numbers in our set, we're essentially dividing these five factors into three groups. It's like saying, "Hey, how can we split these guys up?"

Let's walk through the different ways we can do this. We need to think systematically to make sure we don't miss any possibilities. This systematic approach is super important in combinatorics. Start with one prime factor, then consider the combinations of two and three, and finally consider combinations of four.

Listing the Possibilities: A Systematic Approach

Okay, let’s get our hands dirty and list out the possibilities. This is where we transform from math students to math detectives, and that's seriously cool.

1. One factor in a number, one factor in another, and three factors in the last number:

   • For example, a = 2, b = 3, and c = 5 × 7 × 11 = 385. The possibilities here will be to combine 3 prime factors into one number, leaving the remaining 2 primes to form the other two numbers. The number of ways to choose 3 primes out of 5 is given by the combination formula ⁵C₃ = 5! / (3! * 2!) = 10. These are the most basic groupings, and they set the stage for more complex combinations. Each grouping represents a unique way to distribute our prime factors, ensuring we capture every possible set. This is the core of our systematic exploration.

2. Two factors in one number, two factors in another, and one factor in the last number:

   • For example, a = 2 × 3 = 6, b = 5 × 7 = 35, and c = 11. The number of ways to choose 2 primes out of 5 for the first number is ⁵C₂ = 10. After choosing 2 primes, we need to choose another 2 from the remaining 3 primes, which gives us ³C₂ = 3 ways. However, since the order of choosing the pairs doesn't matter, we divide by 2! (the number of ways to arrange the two pairs), giving us (10 * 3) / 2 = 15 ways. These combinations are a bit more intricate, requiring us to carefully consider how pairs of primes can be formed. It's like we're choreographing a mathematical dance, ensuring each pair complements the others perfectly. Understanding these combinations is crucial for mastering the problem.

Now, let's pause and think. Can we have three numbers each with more than two factors? Nope! If we had something like 2 × 3, 5 × 7, and 11, we've already covered that in our second case. This critical check ensures that we haven't missed any scenarios. This careful consideration of constraints highlights the precision needed in combinatorial problems, and it's a skill that translates well beyond mathematics.

The Final Count: Putting It All Together

So, let's add up the possibilities:

• 10 ways (one number with three factors)
• 15 ways (two numbers with two factors)

Adding these up, we get 10 + 15 = 25 distinct sets.

Therefore, the correct answer is D) 25. We've solved it, guys! This final calculation is the culmination of our hard work, and it's incredibly satisfying to see the solution emerge from the systematic application of our techniques.

Why This Works: The Underlying Principles

You might be wondering, “Okay, we got the answer, but why does this method work?” That’s a fantastic question! It's not enough to just get the solution; we need to understand the 'why' behind it. That's what really makes math click.

The key here is that we used prime factorization to break down the number into its fundamental components. Prime numbers are the atoms of the number world – they are the smallest building blocks that make up all other numbers. By breaking 2310 into its prime factors, we made the problem of finding sets much more manageable. It's like taking a complex machine apart to see how each piece fits together. This principle of breaking down complexity into manageable parts is a recurring theme in mathematics.

Then, we used combinatorics, which is the art of counting combinations and permutations. Combinatorics helps us to systematically explore all the different ways we can group these prime factors. This systematic approach ensures that we don’t miss any possibilities and that we count each distinct set exactly once. Combinatorics provides the tools and techniques to navigate these complex counting scenarios.

Tips for Tackling Similar Problems

So, what if you encounter a similar problem in the future? Here are a few tips to keep in your mathematical toolkit:

1. Prime Factorization is Your Friend: Always start by breaking down the number into its prime factors. This simplifies the problem and makes it easier to see the possible combinations.
2. Be Systematic: List out the possibilities in a systematic way. Don’t just guess and check. A systematic approach ensures you don’t miss any solutions.
3. Think About Combinations: Use combinatorics to help you count the number of ways to group the factors. The combination formula (nCr) is your best friend here.
4. Double-Check: Always double-check your work to make sure you haven’t missed any cases or counted any cases twice. This is super important to reduce errors!

Conclusion

And there you have it! We've successfully navigated a tricky math problem involving distinct sets of natural numbers. We used prime factorization, systematic listing, and a bit of combinatorics to arrive at the answer. More importantly, we understood the underlying principles behind the solution. Remember, math isn't just about getting the right answer; it's about understanding the process and the reasoning behind it. Mastering these principles opens the door to tackling a wide range of mathematical challenges.

So, the next time you encounter a problem like this, don't be intimidated! Break it down, be systematic, and most importantly, have fun with it. Math is a beautiful puzzle, and you've got the tools to solve it. Keep exploring, keep questioning, and keep learning! You guys are awesome, and I know you can conquer any math challenge that comes your way! This final encouragement highlights the importance of perseverance and a positive mindset in mathematics, and it reminds us that problem-solving is not just about the answer, but also about the journey of discovery. Now, go forth and solve some math puzzles!