gcf of 65 and 72

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05-02-2025

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Finding the Greatest Common Factor (GCF) of 65 and 72

Title Tag: GCF of 65 & 72: A Step-by-Step Guide

Meta Description: Learn how to find the greatest common factor (GCF) of 65 and 72 using the prime factorization method and the Euclidean algorithm. This easy-to-follow guide provides step-by-step instructions and examples. Master GCF calculations today!

Understanding Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides two or more integers without leaving a remainder. Finding the GCF is a fundamental concept in mathematics used in various applications, from simplifying fractions to solving algebraic problems. In this article, we'll determine the GCF of 65 and 72 using two common methods.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

1. Find the prime factorization of 65:

   65 = 5 x 13

2. Find the prime factorization of 72:

   72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

3. Identify common prime factors:

   Examining the prime factorizations of 65 and 72, we see they share no common prime factors.

4. Calculate the GCF:

   Since there are no common prime factors, the greatest common factor of 65 and 72 is 1.

Method 2: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the GCF, particularly for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0.

1. Divide the larger number (72) by the smaller number (65):

   72 ÷ 65 = 1 with a remainder of 7

2. Replace the larger number with the smaller number (65) and the smaller number with the remainder (7):

   Now we find the GCF of 65 and 7.

3. Repeat the division:

   65 ÷ 7 = 9 with a remainder of 2

4. Repeat again:

   7 ÷ 2 = 3 with a remainder of 1

5. Repeat one last time:

   2 ÷ 1 = 2 with a remainder of 0

6. The GCF is the last non-zero remainder:

   The last non-zero remainder was 1. Therefore, the GCF of 65 and 72 is 1.

Conclusion

Both the prime factorization method and the Euclidean algorithm confirm that the greatest common factor of 65 and 72 is 1. This means that 65 and 72 are relatively prime; they share no common factors other than 1. Understanding these methods allows you to efficiently determine the GCF of any pair of integers. Remember to choose the method that best suits the numbers you are working with – prime factorization is often easier for smaller numbers, while the Euclidean algorithm is more efficient for larger ones.