gcf of 65 and 72

Charlotte

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27-02-2025

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Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics. This article will guide you through several methods to determine the GCF of 65 and 72. We'll explore the prime factorization method and the Euclidean algorithm, providing a clear understanding of the process.

Understanding Greatest Common Factor (GCF)

The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into all of them. In simpler terms, it's the biggest number that's a factor of both numbers. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides evenly into both 12 and 18.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

Step 1: Find the prime factorization of 65.

65 = 5 x 13

Step 2: Find the prime factorization of 72.

72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

Step 3: Identify common prime factors.

Looking at the prime factorizations of 65 and 72, we see they have no prime factors in common.

Step 4: Determine the GCF.

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

Method 2: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the GCF, especially for larger numbers. It uses repeated division with remainders.

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

72 ÷ 65 = 1 with a remainder of 7.

Step 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.

Step 3: Repeat the division process.

65 ÷ 7 = 9 with a remainder of 2.

Step 4: Continue the process.

7 ÷ 2 = 3 with a remainder of 1.

Step 5: Continue until the remainder is 0.

2 ÷ 1 = 2 with a remainder of 0.

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

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

Conclusion

Using both the prime factorization method and the Euclidean algorithm, we've definitively shown that the greatest common factor of 65 and 72 is 1. This means that 1 is the largest number that divides evenly into both 65 and 72. Understanding these methods allows you to efficiently find the GCF of any pair of numbers.