Introduction to Finding the Greatest Common Divisor (GCD) for Measuring Room Dimensions

When dealing with measurements in a practical context, such as determining the largest tape measure that can exactly measure the three orthogonal dimensions of a room, the concept of the Greatest Common Divisor (GCD) comes in very handy. This article will step through the process of finding the GCD to determine the longest tape that can measure the length, breadth, and height of a room with dimensions 825 cm, 675 cm, and 450 cm, respectively.

Understanding the Problem

Given the dimensions of a room: length 8 m 25 cm, breadth 6 m 75 cm, and height 4 m 50 cm. These dimensions need to be converted to centimeters for easy handling. Thus, the length is 825 cm, the breadth is 675 cm, and the height is 450 cm.

Step 1: Prime Factorization

The first step in finding the GCD is to factorize the three given dimensions into their prime factors.

Prime Factorization

825 825 3 × 52 × 11 675 675 33 × 52 450 450 2 × 32 × 52

Step 2: Identifying Common Factors

Next, we identify the common factors among the three factorizations to find the GCD.

Common Factors

For 3: The minimum power is 31 from 825. For 5: The minimum power is 52 which appears in all. For 2: It appears in 450 but not in the others, so it is not included.

Step 3: Calculating the GCD

Using the identified common factors, we can now calculate the GCD:

Calculation of GCD

[text{GCD} 3^1 times 5^2 3 times 25 75 text{ cm}]

Conclusion and Practical Application

The longest tape that can measure the three dimensions of the room exactly is 75 cm. This method can be applied in various practical situations where it is necessary to find the largest unit that can divide a set of measurements without a remainder, such as in carpentry, interior designing, or construction.

Other Practical Applications

The GCD is a fundamental concept in mathematics that has numerous applications beyond measuring room dimensions. It is used in cryptography, computer science, and various practical scenarios, such as finding the smallest common multiple or simplifying fractions. Knowing the GCD can often simplify calculations and provide accurate results in real-world problems.

Conclusion

In conclusion, the process of finding the GCD using prime factorization and common factor identification is a powerful tool in mathematics. It enables us to determine the longest tape that can measure the exact dimensions of a room, aiding in accurate and efficient measurements.