greatest common factor of 48 and 80

Guri Bee

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18-10-2024

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Finding the Greatest Common Factor (GCD) of 48 and 80: A Step-by-Step Guide

The greatest common factor (GCD) of two numbers is the largest number that divides both numbers evenly. Finding the GCD is a fundamental concept in mathematics with applications in various fields, including number theory and cryptography.

In this article, we'll explore different methods to determine the GCD of 48 and 80, focusing on clarity and understanding.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. The GCD is then the product of all common prime factors raised to the lowest power they appear in either factorization.

1. Prime Factorization of 48: 48 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3¹

2. Prime Factorization of 80: 80 = 2 x 2 x 2 x 2 x 5 = 2⁴ x 5¹

3. Identify Common Prime Factors: Both 48 and 80 share the prime factor 2, with the lowest power being 4 (2⁴).

4. Calculate GCD: The GCD of 48 and 80 is 2⁴ = 16.

Therefore, the greatest common factor of 48 and 80 is 16.

Method 2: Euclidean Algorithm

The Euclidean Algorithm is an efficient method for finding the GCD, especially for larger numbers. It works by repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is zero. The last non-zero remainder is the GCD.

1. Divide the larger number by the smaller number: 80 ÷ 48 = 1 remainder 32

2. Replace the larger number with the smaller number, and the smaller number with the remainder: 48 ÷ 32 = 1 remainder 16

3. Repeat steps 1 and 2 until the remainder is zero: 32 ÷ 16 = 2 remainder 0

Since the remainder is now zero, the last non-zero remainder, 16, is the GCD of 48 and 80.

Practical Applications of GCD

Finding the GCD has practical applications in various areas:

• Simplifying Fractions: The GCD can be used to simplify fractions by dividing both the numerator and denominator by the GCD. For example, the fraction 48/80 can be simplified to 3/5 by dividing both numbers by 16.
• Finding the Least Common Multiple (LCM): The GCD and LCM of two numbers are related by the formula: LCM(a, b) x GCD(a, b) = a x b.
• Cryptography: GCD is used in cryptography algorithms like RSA, where it helps generate keys and encrypt messages.

Conclusion:

Finding the GCD of two numbers is a fundamental mathematical operation with practical applications in various fields. By understanding the methods of prime factorization and the Euclidean Algorithm, you can efficiently determine the GCD of any two integers.