Greatest Common Factor Problems Students In Rows And Drum Capacity

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Hey guys! Today, we're diving into a couple of super interesting math problems that involve finding the greatest common factor (GCF). These types of problems might seem tricky at first, but once you understand the concept, they become a piece of cake. We'll break down each problem step by step, so you'll be solving them like a pro in no time! So, let's jump right in and make math fun!

Problem 1 Finding the Greatest Number of Students in Rows

Understanding the Problem

Our first problem involves a school with three classes: third, fourth, and fifth. There are 154 students in third grade, 126 in fourth grade, and 112 in fifth grade. The question is: what is the greatest number of students that can be arranged in rows so that each row has the same number of students? This is a classic GCF problem! Whenever you see the phrase "greatest number" or something similar, it's a big hint that you need to find the GCF.

The main goal here is to find a number that divides evenly into 154, 126, and 112. Imagine you're trying to organize these students for a school event. You want to make sure each row looks neat and has the same number of students. If you put, say, 10 students in each row, it wouldn't work because 154, 126, and 112 aren't divisible by 10. Some students would be left out, and that's not what we want!

So, how do we find this magic number? That's where the GCF comes in. The greatest common factor is the largest number that divides evenly into two or more numbers. Think of it as the biggest piece of a puzzle that fits perfectly into all the given numbers. To solve this, we'll use a method that breaks down each number into its prime factors. This method is super useful because it helps us see all the little building blocks that make up each number. Once we have the prime factors, we can easily find the GCF. It's like having all the ingredients for a recipe laid out in front of us – then we just need to pick the ones they all share!

Finding the GCF

To find the GCF of 154, 126, and 112, we'll use the prime factorization method. This means we'll break down each number into its prime factors – those numbers that can only be divided by 1 and themselves (like 2, 3, 5, 7, etc.).

Let's start with 154:

  • 154 can be divided by 2: 154 = 2 x 77
  • 77 can be divided by 7: 77 = 7 x 11
  • So, the prime factors of 154 are 2 x 7 x 11

Now, let's do 126:

  • 126 can be divided by 2: 126 = 2 x 63
  • 63 can be divided by 3: 63 = 3 x 21
  • 21 can be divided by 3: 21 = 3 x 7
  • So, the prime factors of 126 are 2 x 3 x 3 x 7 (or 2 x 3² x 7)

Finally, let's break down 112:

  • 112 can be divided by 2: 112 = 2 x 56
  • 56 can be divided by 2: 56 = 2 x 28
  • 28 can be divided by 2: 28 = 2 x 14
  • 14 can be divided by 2: 14 = 2 x 7
  • So, the prime factors of 112 are 2 x 2 x 2 x 2 x 7 (or 2⁴ x 7)

Now that we have the prime factors for each number, we can identify the common factors. These are the prime numbers that appear in all three factorizations. Looking at our lists:

  • 154 = 2 x 7 x 11
  • 126 = 2 x 3 x 3 x 7
  • 112 = 2 x 2 x 2 x 2 x 7

We can see that both 2 and 7 are common factors. To find the GCF, we multiply these common factors together: GCF = 2 x 7 = 14. Therefore, the greatest number of students that can stand in rows with an equal number of students in each row is 14.

Solution and Explanation

The greatest number of students that can be made to stand in rows having an equal number of students in each is 14. This means you can arrange the students from each class into rows of 14 students each, and you won't have any students left over. Let's see how many rows each class would have:

  • Third grade (154 students): 154 / 14 = 11 rows
  • Fourth grade (126 students): 126 / 14 = 9 rows
  • Fifth grade (112 students): 112 / 14 = 8 rows

So, you'd have 11 rows of third graders, 9 rows of fourth graders, and 8 rows of fifth graders. Each row would have 14 students, making the arrangement neat and organized. This is the power of the GCF! It helps us solve real-world problems by finding the largest number that fits perfectly into multiple quantities. Whether you're arranging students, cutting fabric, or even planning a party, understanding the GCF can be super helpful. Remember, it's all about finding that biggest piece that fits into everything!

Problem 2: Finding the Capacity of the Greatest Container

Understanding the Problem

Now, let's move on to our second problem. This time, we're dealing with three drums containing different amounts of liquid: 90 litres, 108 litres, and 144 litres. The question asks us to find the maximum capacity of a container that can measure the liquid in each drum an exact number of times. This is another classic GCF problem, just disguised in a slightly different scenario! When you see "maximum capacity" or "greatest measure", it's a signal that we're back in GCF territory.

Imagine you have these three drums, and you need to pour the liquid from each drum into smaller containers. You want to use the same container for all three drums, and you want to make sure you use the fewest number of containers possible. This means your container needs to be as big as possible while still being able to measure each drum's contents perfectly. If you chose a 1-litre container, you could do it, but you'd need 90 containers for the first drum, 108 for the second, and 144 for the third. That's a lot of containers!

So, we need to find a container size that is a common factor of 90, 108, and 144. But not just any common factor – we want the greatest common factor. This will give us the maximum capacity of the container we can use. Just like in our previous problem, we'll use prime factorization to break down each number and find the GCF. It's like having a set of measuring cups and trying to find the largest one that fits perfectly into all your mixing bowls. Let's get started!

Finding the GCF

To find the GCF of 90, 108, and 144, we'll again use the prime factorization method. This will help us break down each number into its prime factors, making it easier to identify the common factors.

Let's start with 90:

  • 90 can be divided by 2: 90 = 2 x 45
  • 45 can be divided by 3: 45 = 3 x 15
  • 15 can be divided by 3: 15 = 3 x 5
  • So, the prime factors of 90 are 2 x 3 x 3 x 5 (or 2 x 3² x 5)

Now, let's break down 108:

  • 108 can be divided by 2: 108 = 2 x 54
  • 54 can be divided by 2: 54 = 2 x 27
  • 27 can be divided by 3: 27 = 3 x 9
  • 9 can be divided by 3: 9 = 3 x 3
  • So, the prime factors of 108 are 2 x 2 x 3 x 3 x 3 (or 2² x 3³)

Finally, let's factorize 144:

  • 144 can be divided by 2: 144 = 2 x 72
  • 72 can be divided by 2: 72 = 2 x 36
  • 36 can be divided by 2: 36 = 2 x 18
  • 18 can be divided by 2: 18 = 2 x 9
  • 9 can be divided by 3: 9 = 3 x 3
  • So, the prime factors of 144 are 2 x 2 x 2 x 2 x 3 x 3 (or 2⁴ x 3²)

Now that we have the prime factors for each number, let's identify the common factors:

  • 90 = 2 x 3 x 3 x 5
  • 108 = 2 x 2 x 3 x 3 x 3
  • 144 = 2 x 2 x 2 x 2 x 3 x 3

We can see that 2 and 3 appear in all three factorizations. To find the GCF, we multiply the lowest powers of the common factors together. In this case, we have 2¹ (since 2 appears once in the factorization of 90) and 3² (since 3 appears twice in all three factorizations). So, the GCF = 2 x 3 x 3 = 18. Therefore, the maximum capacity of the container is 18 litres.

Solution and Explanation

The maximum capacity of a container that can measure the liquid in each drum an exact number of times is 18 litres. This means you can use an 18-litre container to measure the liquid in each drum without any leftovers. Let's see how many times you'd need to use the container for each drum:

  • 90 litres drum: 90 / 18 = 5 times
  • 108 litres drum: 108 / 18 = 6 times
  • 144 litres drum: 144 / 18 = 8 times

So, you'd need to use the 18-litre container 5 times for the first drum, 6 times for the second, and 8 times for the third. This shows how the GCF helps us find the largest measure that works perfectly for multiple quantities. Whether you're measuring liquids, cutting lengths of rope, or dividing up tasks, the GCF is a super useful tool in many real-life situations. It helps us find the biggest common piece, making things efficient and organized!

Conclusion

So, guys, we've tackled two awesome problems today, all thanks to the power of the Greatest Common Factor (GCF)! We saw how to find the greatest number of students for rows and the maximum capacity of a container. The key takeaway here is that whenever you encounter problems asking for the “greatest” or “maximum” common measure, think GCF!

Remember, the prime factorization method is your best friend for finding the GCF. Break down each number into its prime factors, identify the common factors, and multiply them together (using the lowest powers if needed). Once you get the hang of it, you'll be able to solve these types of problems in no time. Keep practicing, and you'll become a GCF master!

Math might seem like a bunch of abstract rules and formulas, but it's actually super practical and helps us solve real-world problems every day. Understanding concepts like the GCF can make everyday tasks easier and more efficient. So, keep exploring, keep learning, and most importantly, keep having fun with math! You've got this!