Total Points Of Students Based On Divisors

Hey guys! Let's dive into a fun math problem where we need to figure out the total points scored by four students. Each student's score is based on a pretty cool concept: the number of positive integer divisors of their assigned number. Sounds intriguing, right? So, we have Ali, Burcu, Cem, and Deniz, each with their own number. Our mission is to find out how many points each of them scored and then add 'em all up. Buckle up, because we're about to break down each student's score step-by-step!

Ali's Score: Diving into the Divisors of 24

Okay, let's kick things off with Ali. Ali's number is 24, and to figure out his score, we need to find all the positive integer divisors of 24. Now, what exactly are divisors? Well, divisors are numbers that divide evenly into another number, leaving no remainder. So, we need to find all the numbers that divide 24 perfectly.

Let's start listing them out systematically. Of course, 1 is always a divisor, so that's our first one. Then, 2 divides into 24, giving us 12. Next up, 3 works too, because 24 divided by 3 is 8. We can keep going: 4 is a divisor (24 / 4 = 6), and 6 is also a divisor (24 / 6 = 4). We already found 6, so let's keep going. 8 is also one since 24 divided by 8 is 3. And 12, as we've seen, is a divisor. Last but not least, 24 itself is a divisor because any number is divisible by itself.

So, if we list all the divisors, we have: 1, 2, 3, 4, 6, 8, 12, and 24. Count 'em up! That's a total of 8 divisors. This means Ali scores 8 points. See, it's not as complicated as it might seem at first. We just methodically found each number that divides 24 without leaving a remainder. Keep this method in mind, because we'll be using it for the other students too!

Burcu's Score: Unpacking the Divisors of 49

Now, let’s move on to Burcu. Burcu has the number 49, and like with Ali, we need to find all the positive integer divisors of 49 to determine her score. What numbers divide evenly into 49? Well, this one might be a bit simpler than 24, but let's go through it step by step to make sure we don't miss anything.

We always start with 1, because 1 is a divisor of every number. Then, we try 2, but 49 divided by 2 leaves a remainder, so 2 is out. How about 3? Nope, 49 isn’t divisible by 3 either. Let’s keep going. 4 doesn’t work, and neither does 5 or 6. When we get to 7, we hit the jackpot! 49 divided by 7 is exactly 7. So, 7 is a divisor.

Now, here’s a cool thing about divisors: when you find a divisor, you sometimes find a pair. In this case, 7 times 7 equals 49. So, we've kind of already found its "pair." Let's keep going just to be sure we haven't missed anything. 8, 9, 10, and so on, won't work. The next number that divides 49 evenly is 49 itself, because every number is divisible by itself.

So, the positive integer divisors of 49 are: 1, 7, and 49. That’s a total of 3 divisors. So Burcu scores 3 points. Not bad, Burcu! You see, sometimes the numbers we're dealing with have fewer divisors, which can make our job a little easier. Now let's tackle Cem's score!

Cem's Score: Decoding the Divisors of 32

Alright, time to figure out Cem's score. Cem's number is 32, and as you guys know by now, our mission is to find all the positive integer divisors of 32. Let's roll up our sleeves and get to work. We'll use the same method we used for Ali and Burcu: systematically checking each number to see if it divides 32 evenly.

As always, we start with 1. 1 divides into everything, so it’s definitely a divisor of 32. Next, we try 2. 32 divided by 2 is 16, so 2 is a divisor. Things are looking good so far! Let’s keep going. Does 3 divide 32 evenly? Nope, there’s a remainder. How about 4? Yep, 32 divided by 4 is 8, so 4 is a divisor.

Okay, let’s keep going. 5 doesn’t work, nor does 6. But 7 also doesn't do the trick. Now, 8 does the trick, because 32 divided by 8 is 4. We’ve actually already found 8's "pair" when we found 4, but it’s good to keep checking. Let’s continue. 9, 10, 11, 12, 13, 14, and 15 won't work. But wait, 16 is a divisor, because 32 divided by 16 is 2. We found 16's pair earlier too! And finally, 32 is a divisor of itself, because any number divided by itself is 1.

So, the positive integer divisors of 32 are: 1, 2, 4, 8, 16, and 32. Count them up! That's a total of 6 divisors. This means Cem scores 6 points. Awesome job, Cem! Now we have just one more student to go. Let’s see what Deniz has in store for us.

Deniz's Score: Discovering the Divisors of 13

Last but not least, we have Deniz. Deniz's number is 13, and we need to find all the positive integer divisors of 13. Now, this one might be a little trickier, or maybe even easier, depending on how you look at it. Let's dive in and find out!

We always start with 1, because, as we know, 1 is a divisor of every number. Now, let's try 2. 13 divided by 2 leaves a remainder, so 2 is not a divisor. What about 3? Nope, 13 isn't divisible by 3 either. Let’s keep going: 4, 5, 6, 7, 8, 9, 10, 11, and 12 – none of these divide 13 evenly.

Now, what about 13 itself? Well, of course, 13 divided by 13 is 1, so 13 is a divisor. But are there any other divisors? If you try any number between 1 and 13, you'll find that none of them divide 13 without leaving a remainder. This is because 13 is what we call a prime number.

Prime numbers are special because they only have two divisors: 1 and themselves. So, the positive integer divisors of 13 are just 1 and 13. That’s a total of 2 divisors. This means Deniz scores 2 points. Nice one, Deniz! Now that we've figured out everyone's individual scores, we're ready for the final step.

Calculating the Total Score: Adding It All Up

Okay, guys, we've done the hard work of figuring out each student's score. Now comes the satisfying part: adding them all up to find the total score. We have Ali with 8 points, Burcu with 3 points, Cem with 6 points, and Deniz with 2 points. All we need to do is add these numbers together.

So, let's do it: 8 (Ali) + 3 (Burcu) + 6 (Cem) + 2 (Deniz) = ?. Let’s add these numbers step by step. 8 plus 3 is 11. Then, we add 6 to 11, which gives us 17. Finally, we add 2 to 17, and we get 19.

Therefore, the total score of the four students is 19 points. That’s it! We've solved the problem. We broke down each student's score by finding the divisors of their numbers, and then we added those scores together. Great job, everyone!

In conclusion, this problem was a fun way to practice finding divisors and adding numbers together. Remember, math isn't just about formulas and equations; it's also about problem-solving and thinking step-by-step. Keep practicing, and you'll become math whizzes in no time!