The greatest common factor (GCF) of two or more numbers is the largest positive integer that divides each of the given numbers without a remainder. Finding the GCF is an essential mathematical operation used in various applications, including simplifying fractions, solving equations, and working with polynomials. In this article, we will explore the GCF of 36 and 84 in detail, providing a step-by-step explanation, exploring its properties, and covering related concepts.

The GCF of 36 and 84 is the largest positive integer that divides both 36 and 84 evenly. To find the GCF, we can use the prime factorization method. Prime factorization involves breaking down each number into its unique prime factors. For 36, the prime factors are 2 x 2 x 3 x 3, and for 84, the prime factors are 2 x 2 x 3 x 7. The GCF of 36 and 84 is the product of the common prime factors, which is 2 x 2 x 3 = 12.

Therefore, the GCF of 36 and 84 is 12. This means that 12 is the largest positive integer that divides both 36 and 84 without leaving any remainder.

## Prime Factorization

### Definition

Prime factorization is a method for expressing a number as a product of its prime factors. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Prime factorization involves breaking down a number into its smallest prime factors.

### Example

To find the prime factorization of 36, we can start by dividing it by the smallest prime number, 2. 36 divided by 2 is 18, which is also divisible by 2. We continue this process until we reach a prime number. Therefore, the prime factorization of 36 is 2 x 2 x 3 x 3.

### Properties of Prime Factorization

- Every number has a unique prime factorization.
- The GCF of two numbers is the product of their common prime factors.
- The LCM of two numbers is the product of all their prime factors.

## Greatest Common Factor (GCF)

### Definition

The GCF of two or more numbers is the largest positive integer that divides each of the given numbers without a remainder.

### Method of Finding GCF

To find the GCF of two or more numbers, we can use the following steps:

- Prime factorize each number.
- Identify the common prime factors.
- Multiply the common prime factors together.

### Example

To find the GCF of 36 and 84, we follow these steps:

- Prime factorize 36: 2 x 2 x 3 x 3
- Prime factorize 84: 2 x 2 x 3 x 7
- Identify the common prime factors: 2 x 2 x 3
- Multiply the common prime factors together: 2 x 2 x 3 = 12

Therefore, the GCF of 36 and 84 is 12.

## Least Common Multiple (LCM)

### Definition

The LCM of two or more numbers is the smallest positive integer that is divisible by each of the given numbers without a remainder.

### Method of Finding LCM

To find the LCM of two or more numbers, we can use the following steps:

- Prime factorize each number.
- Identify all the prime factors.
- Multiply all the prime factors together.

### Example

To find the LCM of 36 and 84, we follow these steps:

- Prime factorize 36: 2 x 2 x 3 x 3
- Prime factorize 84: 2 x 2 x 3 x 7
- Identify all the prime factors: 2 x 2 x 3 x 3 x 7
- Multiply all the prime factors together: 2 x 2 x 3 x 3 x 7 = 630

Therefore, the LCM of 36 and 84 is 630.

## Applications of GCF and LCM

### Simplifying Fractions

GCF can be used to simplify fractions by dividing both the numerator and denominator by their GCF.

### Solving Equations

GCF can be used to solve equations by factoring out the GCF from both sides of the equation.

### Working with Polynomials

GCF can be used to factor polynomials by factoring out the GCF from each term.

## Conclusion

The GCF of 36 and 84 is 12. Finding the GCF is a fundamental mathematical operation with various applications in different fields. Understanding the concept of GCF and LCM is crucial for simplifying fractions, solving equations, and working with polynomials. Prime factorization is a powerful tool for finding the GCF and LCM of numbers.

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