Unveiling Common Factors: A Math Puzzle Explained

Hey math enthusiasts! Let's dive into a fun problem involving some cool numbers and their shared secrets. We're going to explore the concept of common factors, which are numbers that divide evenly into two or more given numbers. Get ready to flex those math muscles and uncover the factors that these three numbers have in common. We'll break down the numbers, find their factors, and then pinpoint the ones they all share. It's like a treasure hunt, but instead of gold, we're after the building blocks of these numbers! Let's get started, shall we?

Decoding the Numbers: Prime Factorization Power

Our puzzle revolves around three numbers, expressed a bit differently than usual, designed to make you think a bit more creatively. These are (33² - 3²), (8² + 296), and (36 × 35). The real key to unlock this math problem is understanding prime factorization, the process of breaking down a number into a product of prime numbers. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (like 2, 3, 5, 7, and so on). This is how we'll find the factors of our three numbers. Let’s start with the first number, (33² - 3²). It may look scary, but we can simplify it first! Remember, we can use the difference of squares formula, a² - b² = (a+b)(a-b).

Applying this formula, we get (33 + 3)(33 - 3) = 36 × 30. Now, let's break down 36 and 30 into their prime factors. 36 is 2² × 3² (2 times 2 times 3 times 3), and 30 is 2 × 3 × 5 (2 times 3 times 5). Therefore, the first number, (33² - 3²), simplifies to (2² × 3²) × (2 × 3 × 5), or 2³ × 3³ × 5. Nice, we have the first prime factorization ready! Next up is the second number, (8² + 296). This is a bit simpler since we don't need any special formulas. First, calculate 8² which is 64. Next, add 296 to get 360. Okay, let’s find the prime factors of 360. We can start by dividing by 2 to get 180, then by 2 again to get 90, then by 2 one last time to get 45. Now, we divide 45 by 3 to get 15 and by 3 again to get 5. So, the prime factorization of 360 is 2³ × 3² × 5. Awesome, we’re doing great!

Finally, let’s find the prime factors of the third number: (36 × 35). This is super easy; we just need to break down 36 and 35. We already know from the first number, 36 is 2² × 3². 35 is 5 × 7. So, the prime factorization of (36 × 35) is 2² × 3² × 5 × 7. You see how prime factorization is a superpower in disguise? It makes complex numbers easier to understand! Now that we have the prime factorizations for all three numbers, the fun part begins: identifying the factors they all share. Keep in mind that we’re looking for the common elements in all the factorizations to solve our problem. It’s like searching for the common ingredients in three different recipes.

Pinpointing the Common Ground: Shared Factors

Now that we have the prime factorizations of all three numbers, let’s find the factors they share. This is where we compare the prime factors of each number and identify the ones that appear in all three. For our first number, (33² - 3²), we found the prime factorization to be 2³ × 3³ × 5. The second number, (8² + 296), gave us 2³ × 3² × 5. Lastly, the third number, (36 × 35), is 2² × 3² × 5 × 7. We're looking for prime numbers and their exponents that appear in all three factorizations. Let's start with the smallest prime number: 2.

Looking at the prime factorizations, the lowest power of 2 that appears in all three is 2². Similarly, let's check the number 3. The lowest power of 3 that appears in all three factorizations is 3². Next up is the prime number 5. The number 5 is a part of each prime factorization with the exponent 1 (or simply 5). Finally, we have the prime number 7, but it is not a part of the first and the second prime factorizations. So, 7 is not a common factor. Therefore, the common factors are 2², 3², and 5. Multiplying these together gives us 2² × 3² × 5 = 4 × 9 × 5 = 180. These are the building blocks that are shared among all three original numbers.

Now, let's look at the answer choices you provided: a. 2³ × 3³ × 7 b. 2² × 3² × 7 c. 2² × 3² × 5 d. 2 × 3² × 5. According to our calculations and the common prime factors we found, the correct answer is c. 2² × 3² × 5. It includes the lowest powers of the prime numbers (2, 3, and 5) that are present in all three of our original numbers. So, the answer is c. 2² × 3² × 5. Great job, guys! You've successfully navigated the math puzzle.

Wrapping Up: Mastering the Art of Factorization

Alright, folks, we've reached the end of our factorization journey. We've seen how prime factorization helps us understand the structure of numbers and find common factors. By breaking down complex numbers into their prime components, we can easily identify the shared elements. This skill is super useful in all sorts of math problems. We used the difference of squares formula and calculations to simplify our numbers and then used prime factorization.

The ability to find common factors is fundamental in various areas of mathematics, from simplifying fractions to understanding the relationships between different numbers. This problem helped us sharpen our skills in prime factorization and apply it to a practical scenario. It also showed us how to approach complex expressions step by step. Remember, practice is key. The more you work with numbers and their factors, the better you'll become at recognizing patterns and solving problems. Keep exploring, keep questioning, and most importantly, keep having fun with math! Hopefully, this explanation has been clear and helpful. Keep up the amazing work, and don't hesitate to practice these concepts with other numbers to solidify your understanding. Until next time, keep exploring the world of math! We hope you enjoyed this journey into the world of factors.