A piece of paper measuring 18 cm×20 cm must be cut into square paper tokens. What is the minimum number of paper tokens that can be cut without waste?

Because there should be no waste, the edge of the paper token should divide the length and width of the piece of paper exactly. Here we have a case of the Greatest Common Factor (GCF). So, you must find Greatest Common Factor whenever you are asked to find a number that completely divides more than one number.

Before we learn Greatest Common Factor, we will try to understand what is a factor.

## What is a Factor?

A factor is any number that divides another number by itself with no remainder. To put it another way, if multiplying two whole numbers results in the making of another number, the numbers we are multiplying are factors of the resulted number because the product is divisible by them.

Example:

Find the factors of the following numbers.

a) 12

b) 20

Now we will move on to learn what is a common factor.

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## What is a Common Factor?

When two or more numbers have the same factor, that factor is referred to as a common factor for those numbers.

Example:

Find the common factors of 12 and 20.

**Solution**:

Therefore, common factors of 12 and 20 are 1, 2, and 4.

Now, you are ready to learn Greatest Common Factor (GCF).

## What Does GCF Mean for Math?

The Greatest Common Factor (GCF) is the largest factor among all the common factors of two or more numbers. The Greatest Common Factor (GCF) is also known as Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

**Example**:

Find the Greatest Common Factor of 12 and 20.

**Solution**:

Common factors of 12 and 20 are 1, 2, and 4.

Since 1<2<4,

The largest factor is 4.

Therefore, the Greatest Common Factor of 12 and 20 is 4.

How to find Greatest Common Factor?

There are three common methods to find the greatest common factor for two or more numbers. They are,

- Listing common factors method
- Prime factorization method
- Long division method

The listing common factors method is more suitable for small numbers.

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### Listing common factors method

This approach makes it simple to identify the common factors between the numbers by listing the factors that each number has. By highlighting the common factors, we can select the greatest one from among all of them.

**Steps**:

- Make a list of the factors of each number.
- Mark each of the common factors.
- Select the largest common factor.

**Example**:

Find the greatest common factor of 12, 16, and 32.

## Prime factorization method

The prime factorization method is another method to find the Greatest Common Factor. In this method, we represent the numbers using the prime factorization and combine the common prime factors.

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## What are prime numbers?

Prime numbers are whole numbers greater than one with only two factors – themselves and one. 1 is not considered a prime number.

**Example**:

Write all prime numbers below 15.

**Solution**:

2, 3, 5, 7, 11, 13

## What is prime factorization?

Prime factorization is showing a number as a product of its prime numbers. You can find prime factors by following way.

- Divide the given number by the smallest prime number and leaves no remainder.
- Divide the result by the smallest prime number again, leaving no remainder.
- Continue the process until you get 1 as a result.
- In the end, multiply each prime factor.

**Example**:

Do prime factorization for the following numbers.

a)66

b)120

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## How to Find the GCF Using Prime Factorization?

The process of selecting common prime factors for two or more numbers and multiplying them is known as combining common prime factors. The result of this process is the Greatest Common Factor.

**Example**:

Find the Greatest Common Factor of 66 and 120 using the prime factorization method.

**Solution**:

As we already calculated,

**Example**:

Find the Greatest Common Factor of 132 and 254 using the prime factorization method.

**Solution**:

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## Long Division Method

For large numbers, it is not very convenient to find the Greatest Common Factor (GCF) by prime factorization. The long division method is more useful when dealing with large numbers to find the GCF.

With this approach, we divide the larger number by the smaller number first. The remainder uses as the new divisor, while the previous divisor acts as the new dividend. We repeat the procedure until there is no longer any remainder. The Greatest Common Factor (GCF) is the divisor that does not leave a remainder.

**Example**:

Find the Greatest Common Factor of 66 and 120 using the long division method.

**Solution**:

Since, 120>66,

120 is the larger number and 66 is the smaller number. Therefore, we need to divide 120 by 66.

As per the above calculation, the Greatest Common Factor of 66 and 120 is 6.

## What is the Greatest Common Factor (GCF) of several algebraic terms?

As the first step, the coefficient of each algebraic term is written as a product of its prime factors and the unknowns are separated and written as a product. Then, we have to find the product of the factors which are common to all algebraic terms.

**Example**:

Find the Greatest Common Factor (GCF) of 4a, 6ab and 8abc.

The common factors of all three algebraic terms, 4a, 6ab and 8abc are 2 and a.

Therefore,

The Greatest Common Factor (GCF)of 4a,6ab,and 8abc=2×a

=2a

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## The relationship between the Greatest Common Factor (GCF) and the Least Common Multiple (LCM)

The smallest common multiple of the given numbers that can be divided exactly by the given numbers without leaving a remainder is known as the LCM (least common multiple). There is one very important relationship between the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). The product of GCF and LCM of two numbers is equal to the product of the numbers.

## Greatest Common Factor (GCF) Examples

**Example**:

Find the GCF of 15, 24, and 42 using the listing factor method.

**Solution**:

1 and 3 are common factors of 15, 24, and 42.

Since 1<3,

The Greatest Common Factor of 15, 24, and 42 is 3.

**Example**:

Find the Greatest Common Factor of 120, 252 and 300 using the prime factorization method.

**Solution**:

**Example**:

Find the Greatest Common Factor of 7,200 and 666 using the long division method.

**Solution**:

Since, 7200>666,

7,200 is the larger number and 666 is the smaller number. Therefore, we need to divide 7,200 by 666.

As per the above calculation, the Greatest Common Factor of 7,200 and 666 is 18.

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## FAQs on Greatest Common Factors

**What is Meant by Greatest Common Factor (GCF)?**

The Greatest Common Factor (GCF) is the largest factor among all the common factors of two or more numbers. The Greatest Common Factor (GCF) is also known as Highest Common Factor (HCF), and Greatest Common Divisor (GCD).

**How to Find the Greatest Common Factor (GCF)?**

There are three common methods to find the greatest common factor for two or more numbers. They are,

· Listing common factors method

· Prime factorization method

· Long division method

The listing common factors method is more suitable for small numbers. The long division method is more useful when dealing with large numbers to find the GCF.

**What is the Greatest Common Factor of Two Prime Numbers?**

A prime number is made up of only two factors (1 and itself). As a result, two prime numbers cannot share any factor other than 1. As a result, the greatest common factor between two prime numbers is always 1.

Example:

Find the GCF of 11 and 19.

Solution:

11=1×11

19=1×19

Therefore, the Greatest Common Factor of 11 and 19 is 1.

**How to Find the Greatest Common Factor of a Polynomial?**

As the first step, the coefficient of each algebraic term is written as a product of its prime factors and the unknowns are separated and written as a product. Then, we have to find the product of the factors which are common to all algebraic terms.

**How do you calculate the GCF using the prime factorization method?**

Present all the numbers using the prime factorization and combine the common prime factors.