Question

For Exercises, solve. A landscaper is planning a sprinkler system that will require precut PVC pipe to be connected and placed in the ground. The trenches he will dig are to be 40 feet, 32 feet, and 24 feet in length. What is the longest length of PVC pipe he can use that will fit the length of each trench exactly and not require any further cutting? How many of the precut pipes will he use in each trench?

Short answer

Answer: The longest length of PVC pipe that can fit each trench exactly is 8 feet. The landscaper will use 5 precut pipes in the 40-foot trench, 4 precut pipes in the 32-foot trench, and 3 precut pipes in the 24-foot trench.

Step-by-step solution

Find the greatest common divisor (GCD) of the trench lengths

To determine the longest PVC pipe length that can fit the length of each trench exactly, we need to find the GCD of 40 feet, 32 feet, and 24 feet. The prime factorization of the numbers is as follows: 40 = 2^3 * 5 32 = 2^5 24 = 2^3 * 3 The GCD is the product of the least powers of all common prime factors. In this case, the common prime factor is 2 and the least power is 2^3 = 8. Therefore, the GCD of these three numbers is 8.

Calculate the number of precut pipes for each trench

Now that we have determined that the longest length of PVC pipe that can fit the length of each trench exactly is 8 feet, we need to calculate how many of these precut pipes will be used in each trench. To do this, we will divide the length of each trench by the GCD (8 feet). For the 40-foot trench: 40 feet / 8 feet = 5 precut pipes For the 32-foot trench: 32 feet / 8 feet = 4 precut pipes For the 24-foot trench: 24 feet / 8 feet = 3 precut pipes So, the landscaper will use 5 precut pipes in the 40-foot trench, 4 precut pipes in the 32-foot trench, and 3 precut pipes in the 24-foot trench.

Key concepts

Understanding the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the greatest common factor, is a key mathematical concept that represents the largest number that can divide two or more numbers without leaving a remainder. This concept is particularly useful in various real-life applications, such as simplifying fractions or determining the optimal amounts of materials needed for construction projects, just like in the example of the landscaper planning a sprinkler system.

When looking for the GCD of a set of numbers, you might start by listing out all the divisors of each number and then identifying the largest one they all share. However, this can be time-consuming for larger numbers. A more efficient technique involves prime factorization, which is the process of breaking down numbers into their basic building blocks - prime numbers. By comparing the prime factors of each number and choosing the smallest powers they have in common, you can easily find the GCD.

For instance, identifying the prime factors of 40, 32, and 24, we find that they all include the prime number 2, but with different powers. Since the objective is to discover the longest piece of PVC pipe that can fit each trench without further cutting, the GCD corresponds to the least power of 2 common to all three numbers, which is 2 cubed or 8. This means that the longest single piece of precut pipe the landscaper can use is 8 feet long.

Prime Factorization

Prime factorization is an invaluable method when calculating the greatest common divisor among several numbers. Let's delve deeper into this process, which disassembles composite numbers into a product of prime factors. A prime number, for reference, is a natural number greater than 1 that has no divisors other than 1 and itself.

How does prime factorization work?

Starting with the smallest prime number, which is 2, divide your number and see if you can get a whole number as a quotient. If you can, 2 is a prime factor. Continue dividing by 2 until you can no longer get a whole number. Move on to the next prime numbers (3, 5, 7, and so on) using the same process, until the final quotient is a prime number itself.

Returning to our landscaping scenario:

• 40 breaks down into prime factors of 2 three times (2 to the power of 3) and 5 once.
• 32 breaks down into 2 five times (2 to the power of 5).
• 24 breaks down into 2 three times (2 to the power of 3) and 3 once.

Understanding this breakdown enables us to navigate to the GCD with precision and efficiency.

Division

Division is one of the basic operations in arithmetic, and it's essential when we need to distribute a number into equal parts. It helps us to figure out how many times one number is contained within another — like how many pipes the landscaper can use for each trench. Division is represented by the symbol '÷' or with a slash '/'.

Applying division in our example:

With the knowledge that the GCD is 8 feet, we can use division to solve the second part of the problem. By dividing the length of each trench by the GCD, we're essentially asking, 'How many 8-foot pipes can we fit into each trench exactly?'

Here's how this works out:

• The 40-foot trench divided by 8 feet gives us 5, so 5 precut pipes fit perfectly.
• Dividing the 32-foot trench by 8 feet results in 4, meaning 4 precut pipes are needed.
• Lastly, the 24-foot trench divided by 8 feet equals 3, so 3 precut pipes are required.

Understanding division allows us to efficiently allocate resources and complete this landscaping project with no waste or need for further modification.