Not using the Euclidean algorithm

Oct 13, 2025 by miodeya.com 3 views

Introduction

As a carpenter, it's essential to make the most out of the materials available to ensure efficient use of resources and minimize waste. In this scenario, we have three wooden planks of lengths 54 cm, 72 cm, and 90 cm. The goal is to determine the maximum length that can be cut from these planks with no wastage. This problem can be approached using mathematical techniques, specifically the concept of greatest common divisor (GCD).

Understanding the Problem

To find the maximum length that can be cut from the planks, we need to consider the factors of each plank's length. The factors of a number are the numbers that can be multiplied together to give the original number. For example, the factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

Factors of Each Plank's Length

54 cm Plank

Factors: 1, 2, 3, 6, 9, 18, 27, 54

72 cm Plank

Factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

90 cm Plank

Factors: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Finding the Greatest Common Divisor (GCD)

The GCD of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. In this case, we need to find the GCD of the three plank lengths. We can use the Euclidean algorithm to find the GCD.

Step 1: Find the GCD of 54 and 72

Using the Euclidean algorithm:

54 = 72 × 0 + 54 72 = 54 × 1 + 18 54 = 18 × 3 + 0

The GCD of 54 and 72 is 18.

Step 2: Find the GCD of 18 and 90

Using the Euclidean algorithm:

90 = 18 × 5 + 0

The GCD of 18 and 90 is 18.

Conclusion

The maximum length that can be cut from the planks with no wastage is the GCD of the three plank lengths, which is 18 cm. This means that the carpenter can cut the planks into pieces of length 18 cm without any waste.

Real-World Applications

This problem has real-world applications in various fields, such as:

Construction: In construction, it's essential to minimize waste and make the most out of materials. This problem can be applied to find the maximum length of lumber that can be cut from a plank without any waste.
Manufacturing: In manufacturing, it's crucial to optimize production processes to minimize waste and maximize efficiency. This problem can be applied to find the maximum length of a product that can be cut from a raw material without any waste.
Engineering: In engineering, it's essential to design systems that minimize waste and maximize efficiency. This problem can be applied to find the maximum length of a component that can be cut from a raw material without any waste.

Conclusion

Q: What is the greatest common divisor (GCD) and how is it used in this problem?

A: The greatest common divisor (GCD) is the largest number that divides each of the numbers without leaving a remainder. In this problem, we use the GCD to find the maximum length that can be cut from the planks with no wastage.

Q: Why is it important to find the GCD in this problem?

A: Finding the GCD is essential in this problem because it allows us to determine the maximum length that can be cut from the planks without any waste. By finding the GCD, we can ensure that we are making the most out of the materials available.

Q: Can the GCD be used in other problems related to cutting materials?

A: Yes, the GCD can be used in other problems related to cutting materials. For example, if we have multiple planks of different lengths and we want to cut them into pieces of a specific length, we can use the GCD to determine the maximum length that can be cut from each plank.

Q: What are some real-world applications of the GCD in cutting materials?

A: Some real-world applications of the GCD in cutting materials include:

Construction: In construction, the GCD can be used to determine the maximum length of lumber that can be cut from a plank without any waste.
Manufacturing: In manufacturing, the GCD can be used to determine the maximum length of a product that can be cut from a raw material without any waste.
Engineering: In engineering, the GCD can be used to determine the maximum length of a component that can be cut from a raw material without any waste.

Q: How can the GCD be calculated?

A: The GCD can be calculated using the Euclidean algorithm, which is a step-by-step process for finding the GCD of two or more numbers.

Q: What are some common mistakes to avoid when calculating the GCD?

A: Some common mistakes to avoid when calculating the GCD include:

Not using the Euclidean algorithm: The Euclidean algorithm is a step-by-step process for finding the GCD of two or more numbers. Not using this algorithm can lead to incorrect results.
Not checking for common factors: When calculating the GCD, it's essential to check for common factors between the numbers. Failing to do so can lead to incorrect results.

Q: Can the GCD be used in other mathematical problems?

A: Yes, the GCD can be used in other mathematical problems. For example, the GCD can be used to solve problems related to fractions, decimals, and percentages.

Q: What are some benefits of using the GCD in mathematical problems?

A: Some benefits of using the GCD in mathematical problems include:

Simplifying complex problems: The GCD can be used to simplify complex problems by breaking them down into smaller, more manageable parts.
Finding common ground: The GCD can be used to find common ground between two or more numbers, which can be useful in a variety of mathematical problems.

Conclusion

In conclusion, the GCD is a powerful mathematical concept that can be used to solve a variety of problems related to cutting materials. By understanding how to calculate the GCD and avoiding common mistakes, we can ensure that we are making the most out of the materials available and solving problems efficiently.