Prime Factors: Which Rope Color Has The Largest Sum?
Hey guys! Let's dive into a fun math problem today where we're figuring out which colored rope has the largest sum of its distinct prime factors. We've got four colors to work with: red (35), yellow (40), blue (28), and pink (18). This isn't just about picking a pretty color; we need to break down these numbers into their prime factors and add 'em up. So, grab your thinking caps, and let's get started!
Breaking Down the Numbers: Prime Factorization
Okay, so prime factorization is the name of the game here. What exactly is that? Simply put, it's like taking a number and breaking it down into its prime number building blocks. Remember, a prime number is a number greater than 1 that has only two factors: 1 and itself (think 2, 3, 5, 7, 11, and so on). Our mission is to express each rope number as a product of these prime numbers. Why is this important? Because we need those distinct prime factors to calculate the sum later. This means that if a prime factor appears more than once, we only count it once when adding them up. Sounds like a plan? Let's break down the numbers for each rope color:
First up, we've got the Red Rope with a number 35. To find the prime factors of 35, we think: what prime numbers can divide 35 evenly? Well, 35 is divisible by 5, which is a prime number. When we divide 35 by 5, we get 7, which is also a prime number. So, the prime factorization of 35 is 5 x 7. See how we broke it down into those prime building blocks? Now, keep these prime factors in mind because they'll be crucial when we calculate the sum later. Remember, we are looking for the sum of distinct prime factors, so we only consider each prime number once, even if it appears multiple times in the factorization.
Next, let's tackle the Yellow Rope with a number 40. This one's a bit bigger, but don't worry, we'll break it down step by step. We can start by dividing 40 by the smallest prime number, which is 2. 40 divided by 2 is 20. So, we know 2 is one of our prime factors. Now, we need to break down 20 further. 20 is also divisible by 2, and 20 divided by 2 is 10. So, we've got another 2 in the mix! We continue with 10, which is also divisible by 2, giving us 5. Now, 5 is a prime number, so we can't break it down any further. Putting it all together, the prime factorization of 40 is 2 x 2 x 2 x 5, or 2³ x 5. Notice how the prime factor 2 appears multiple times? But remember, when we calculate the sum of distinct prime factors, we'll only count the 2 once.
Moving on, we have the Blue Rope with the number 28. Let's see what prime factors make up 28. Just like before, we can start by dividing by the smallest prime number, 2. 28 divided by 2 is 14. So, 2 is definitely a prime factor. Now, let's look at 14. Guess what? It's also divisible by 2! 14 divided by 2 is 7. And 7, you guessed it, is a prime number. So, we've reached the end of the line for 28. The prime factorization of 28 is 2 x 2 x 7, or 2² x 7. Again, we have a prime factor (2) that appears more than once, but we'll only count it once when we sum the distinct prime factors.
Last but not least, we've got the Pink Rope with the number 18. Let's find those prime factors! We can start by dividing 18 by 2, which gives us 9. So, 2 is one of our prime factors. Now, we need to break down 9. 9 isn't divisible by 2, but it is divisible by 3! 9 divided by 3 is 3. And 3 is a prime number, so we're done. The prime factorization of 18 is 2 x 3 x 3, or 2 x 3². Remember the rule: when we add the distinct prime factors, we'll only include each prime number once, even if it appears multiple times in the factorization.
Calculating the Sum of Distinct Prime Factors
Alright, now that we've broken down each number into its prime factors, it's time for the next step: calculating the sum of the distinct prime factors for each rope color. Remember, "distinct" means we only count each unique prime factor once, even if it shows up multiple times in the prime factorization. This is where the prime factorizations we found earlier come into play. We're going to go through each rope color and add up those distinct primes. This step is super important because it will directly lead us to figuring out which rope color has the largest sum, which is our ultimate goal. So, let's roll up our sleeves and get adding!
Let's begin with the Red Rope. We already found that the prime factorization of 35 is 5 x 7. The distinct prime factors are simply 5 and 7. Now, we just need to add them together: 5 + 7 = 12. So, the sum of the distinct prime factors for the Red Rope is 12. This is our baseline, and we'll compare the other rope colors to this sum. Remember this number, as we will need it to determine which rope color has the highest sum of its distinct prime factors. We're on our way to solving this puzzle!
Now, let's move on to the Yellow Rope. The prime factorization of 40 is 2 x 2 x 2 x 5, or 2³ x 5. Remember, we only consider distinct prime factors. So, we have 2 and 5. Let's add 'em up: 2 + 5 = 7. So, the sum of the distinct prime factors for the Yellow Rope is 7. Comparing this to the Red Rope (which had a sum of 12), we can already see that the Yellow Rope has a smaller sum. But we're not done yet! We still need to calculate the sums for the Blue and Pink ropes before we can definitively say which rope has the largest sum.
Next up is the Blue Rope. When we did the prime factorization of 28, we found that it's 2 x 2 x 7, or 2² x 7. Again, we're only interested in the distinct prime factors, which are 2 and 7. Time to add them together: 2 + 7 = 9. So, the sum of the distinct prime factors for the Blue Rope is 9. This is more than the Yellow Rope (which had a sum of 7) but still less than the Red Rope (which had a sum of 12). We're getting closer to our answer, but we still have one more rope color to consider: the Pink Rope.
Finally, let's calculate the sum for the Pink Rope. We determined that the prime factorization of 18 is 2 x 3 x 3, or 2 x 3². The distinct prime factors are 2 and 3. Now, let's add them up: 2 + 3 = 5. So, the sum of the distinct prime factors for the Pink Rope is 5. This is the smallest sum we've seen so far. We've now calculated the sum of the distinct prime factors for all four rope colors: Red (12), Yellow (7), Blue (9), and Pink (5). It's time to compare these sums and figure out which rope color has the largest one.
Finding the Largest Sum: The Verdict
Okay, guys, we've reached the exciting part where we figure out the final answer! We've done all the hard work of prime factorization and calculating the sums of distinct prime factors for each rope color. Now, it's time to put those numbers side-by-side and see which one comes out on top. We're looking for the largest sum, which will tell us which rope color wins this prime factor challenge.
Let's recap the sums we calculated:
- Red Rope: 12
- Yellow Rope: 7
- Blue Rope: 9
- Pink Rope: 5
Looking at these numbers, it's pretty clear which one is the biggest. The Red Rope has a sum of 12, which is larger than all the other sums. Yellow has 7, Blue has 9, and Pink has 5. So, 12 is our winner! This means that the Red Rope has the largest sum of its distinct prime factors compared to the other ropes. We successfully found the solution through our methodical prime factorization and addition.
Conclusion: The Red Rope Reigns Supreme
So, there you have it! We've cracked the code and figured out that the Red Rope is the winner, boasting the largest sum of distinct prime factors. We took on this challenge by first understanding what prime factorization is and then methodically breaking down each rope number into its prime building blocks. Next, we carefully added up the distinct prime factors for each color, making sure to only count each unique prime number once. Finally, we compared those sums and found that the Red Rope's sum of 12 was the highest.
This problem was a fun way to exercise our math skills, especially prime factorization and addition. Remember, these skills aren't just for math problems; they can help you in many areas, from understanding computer algorithms to even figuring out patterns in nature. Keep practicing, and you'll become a prime factorization pro in no time! Great job, everyone, for sticking with it and solving this math puzzle with me. Until next time, keep those numbers crunching!
