what is the gcf of 71 and 82

Natashya

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23-12-2024

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The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides each of the integers without leaving a remainder. Finding the GCF is a fundamental concept in number theory and has applications in various mathematical areas. Let's determine the GCF of 71 and 82.

Understanding Prime Factorization

One common method for finding the GCF involves prime factorization. Prime factorization is the process of breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves.

• Prime Factorization of 71: 71 is a prime number. This means its only factors are 1 and 71. Therefore, the prime factorization of 71 is simply 71.

• Prime Factorization of 82: 82 is an even number, so it's divisible by 2. 82 = 2 x 41. Since 41 is also a prime number, the prime factorization of 82 is 2 x 41.

Finding the GCF Using Prime Factorization

Once we have the prime factorization of both numbers, we identify the common prime factors and their lowest powers. In this case:

• 71: 71
• 82: 2 x 41

There are no common prime factors between 71 and 82. This means their greatest common factor is 1.

The Euclidean Algorithm: An Alternative Method

Another efficient way to find the GCF is using the Euclidean algorithm. This iterative method uses repeated division until the remainder is 0. The last non-zero remainder is the GCF.

1. Divide the larger number (82) by the smaller number (71): 82 ÷ 71 = 1 with a remainder of 11.

2. Replace the larger number with the smaller number (71) and the smaller number with the remainder (11): 71 ÷ 11 = 6 with a remainder of 5.

3. Repeat the process: 11 ÷ 5 = 2 with a remainder of 1.

4. Continue: 5 ÷ 1 = 5 with a remainder of 0.

The last non-zero remainder is 1. Therefore, the GCF of 71 and 82 is 1.

Conclusion: The GCF of 71 and 82 is 1

Both the prime factorization method and the Euclidean algorithm confirm that the greatest common factor of 71 and 82 is 1. This means that 71 and 82 are relatively prime; they share no common factors other than 1. Understanding how to find the GCF is a valuable skill in various mathematical applications.