The greatest common factor (GCF) of two or more numbers is the largest positive integer that is a factor of all the numbers. For example, the GCF of 12 and 36 is 12, since 12 is the largest positive integer that is a factor of both 12 and 36.

Finding the GCF of two numbers is a common problem in mathematics, and there are a number of different methods that can be used to do this. One method is to use the prime factorization of the numbers. The prime factorization of a number is the product of the prime numbers that divide the number. For example, the prime factorization of 12 is 2^2 * 3, and the prime factorization of 36 is 2^2 * 3^2.

To find the GCF of two numbers using their prime factorizations, you simply need to find the prime factors that are common to both numbers. The product of these common prime factors is the GCF.

## Prime Factorization Method

### Example: Finding the GCF of 12 and 36

The prime factorization of 12 is 2^2 * 3, and the prime factorization of 36 is 2^2 * 3^2. The common prime factors are 2^2 and 3, so the GCF of 12 and 36 is 2^2 * 3 = 12.

### Another Example: Finding the GCF of 24 and 42

The prime factorization of 24 is 2^3 * 3, and the prime factorization of 42 is 2 * 3 * 7. The only common prime factor is 2, so the GCF of 24 and 42 is 2.

## Euclidean Algorithm

### Algorithm:

- Let a and b be the two numbers for which you want to find the GCF.
- While b is not equal to 0, do the following:
- Let r = a mod b.
- Set a = b.
- Set b = r.
- Return a.

### Example: Finding the GCF of 12 and 36 Using the Euclidean Algorithm

- Let a = 12 and b = 36.
- 12 mod 36 = 12, so let r = 12, a = 36, and b = 12.
- 36 mod 12 = 0, so the algorithm terminates and returns a = 12.

## Applications of GCF

The GCF of two or more numbers has a number of applications in mathematics, including:

**Simplifying fractions:**The GCF of the numerator and denominator of a fraction can be used to simplify the fraction.**Solving equations:**The GCF of the coefficients of the terms in an equation can be used to simplify the equation and solve it.**Finding common multiples:**The GCF of two or more numbers can be used to find the least common multiple of the numbers.

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