Table of contents:

● What is Greatest Common Factor?

● How We Can Find the GCF of Two Numbers?
● LCM Versus GCF
● Real-life Applications of Greatest Common Factor
● Solved Examples
● Practice problems
● Frequently Asked Questions

What is Greatest Common Factor?

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor/Greatest Common
Denominator (GCD) or Highest Common Factor (HCF) of two or more integers is the largest positive integer
that divides each of the numbers without leaving a remainder. In other words, it is the greatest number that is a
common factor of the given numbers.

For example, consider the numbers 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are
1, 2, 3, 6, 9, and 18. The common factors are 1, 2, 3, and 6. Among these, the greatest common factor is 6.
Therefore, GCF(12, 18) = 6.

The GCF is often denoted as GCF(a, b), where "a" and "b" are the numbers for which we want to find the
greatest common factor.

The GCF is a fundamental concept in number theory and has various applications in mathematics, including
simplifying fractions, factoring polynomials, and solving equations. It helps identify the largest factor that two or
more numbers share, making it a useful tool in many mathematical problems.

How We Can Find the GCF of Two Numbers?

Finding the Greatest Common Factor (GCF) involves identifying the largest positive integer that divides two or
more numbers without leaving a remainder. Here's a step-by-step guide on how to find the GCF:

Method 1: Listing Factors

Step 1. List the factors of each number:

- Identify all the positive integers that evenly divide each of the given numbers.

Step 2. Find the common factors:

- Determine the common factors that both numbers share.

Step 3. Identify the greatest common factor:

- The GCF is the largest among the common factors.

Example: Find the GCF of 24 and 36.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors: 1, 2, 3, 4, 6, 12
GCF(24, 36) = 12

Method 2: Prime Factorization

Step 1. Find the prime factorization of each number:

- Express each number as a product of prime numbers.

Step 2. Identify common prime factors:

- Determine the prime factors that both numbers share.

Step 3. Multiply the common prime factors:

- Multiply the common prime factors to find the GCF.

Example: Find the GCF of 48 and 60.

- Prime factorization of 48: \(2^4 \times 3^1\)

- Prime factorization of 60: \(2^2 \times 3^1 \times 5^1\)

Common prime factors: \(2^2 \times 3^1\)

GCF(48, 60) = \(2^2 \times 3^1 = 12\)

Both methods will give you the same result. Choose the method that you find most comfortable or efficient for
a particular set of numbers.

Method 3: Division Method

Finding the Greatest Common Factor (GCF) using the division method involves repeated division until the
remainder is zero. Here's a step-by-step guide on how to find the GCF using this method:

Step 1: Choose Two Numbers

- Select the two numbers for which you want to find the GCF.

Step 2: Divide the Larger Number by the Smaller Number

- Divide the larger number by the smaller number. Record the quotient and remainder.

Step 3: Replace the Larger Number

- Replace the larger number with the smaller number, and the smaller number with the remainder obtained in
the previous step.

Step 4: Repeat Steps 2 and 3

- Continue the process of dividing the new larger number by the new smaller number until the remainder is
zero. The last non-zero remainder is the GCF.

Example: Find the GCF of 48 and 60.

1. \(60 \div 48 = 1\) with a remainder of 12. Replace 60 with 48 and 48 with 12.
2. \(48 \div 12 = 4\) with no remainder. The GCF is the last non-zero remainder, which is 12.

Therefore, GCF(48, 60) = 12.

This method is also known as the Euclidean Algorithm and is a systematic way to find the GCF of two
numbers. It is efficient and commonly used in practice.

Important Note: The GCF of a set of prime numbers will always be 1 because two or more prime numbers
have no common factor other than 1.

LCM Versus GCF

The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) are two different concepts in
number theory, and they serve different purposes in mathematics. As GCF and LCM are often confused lets
look at them individually.

1. Greatest Common Factor (GCF):

The GCF of two or more numbers is the largest positive integer that divides each of the numbers without
leaving a remainder.
For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without a
remainder (12 ÷ 6 = 2, 18 ÷ 6 = 3).
The GCF is useful for simplifying fractions, factoring polynomials, and solving equations.

2. Least Common Multiple (LCM):

The LCM of two or more numbers is the smallest positive integer that is a multiple of each of the numbers.
For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that is divisible by both 4 and 6 (4 ×
3 = 12, 6 × 2 = 12).
The LCM is used in problems involving multiple cycles or periods, such as finding a common denominator for
fractions.

Key Differences:

- The GCF is concerned with finding the largest common factor of numbers, while the LCM is concerned with
finding the smallest common multiple.
- The GCF is a divisor of the given numbers, while the LCM is a multiple of the given numbers.
- The GCF is used in simplifying expressions, while the LCM is used in finding a common base in various
situations.

In summary, while the GCF focuses on common factors, the LCM focuses on common multiples. Both
concepts are important in different mathematical contexts and problem-solving scenarios.

Real-life Applications of Greatest Common Factor

The concept of the Greatest Common Factor (GCF) has various real-life applications in different fields. Here
are some examples:

1. Sharing Resources: In resource allocation or distribution scenarios, finding the GCF can help determine
the largest quantity that can be evenly distributed among different groups or individuals.

2. Design and Proportions: In design and architecture, finding the GCF is crucial for maintaining proportions
and ensuring that elements are scaled appropriately. This is particularly important in fields like graphic design,
where ratios play a significant role.
3. Manufacturing and Packaging: Industries that involve manufacturing and packaging often use the GCF to
determine the most efficient way to package products. It helps find the largest common size or quantity that fits
into various packaging options.

4. Time Management: When planning schedules or coordinating activities, finding the GCF of different time
intervals can help identify the best times for common activities or events. This is particularly useful in event
planning and project management.

5. Programming and Algorithms: In computer science, algorithms often involve finding the GCF to optimize
processes and improve efficiency. For example, in scheduling tasks or allocating resources in software
development.

6. Utilities and Infrastructure: In urban planning, the GCF may be used to determine the most efficient
placement of utilities such as streetlights or waste bins, ensuring an even distribution throughout a
neighborhood.

7. Mathematics Education: Teaching and learning about factors, multiples, and the GCF are fundamental in
mathematics education. Understanding these concepts helps students solve problems and apply mathematical
reasoning in various situations.

Solved Examples
Example 1. Find the Greatest Common Factor (GCF) of 35, 40, and 100 using the listing factors
method.

Solution:
Let's list the factors of 35, 40, and 100:

● Factors of 35: 1, 5, 7, 35
● Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
● Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

Now, let's identify the common factors:

Common factors: 1, 5

Therefore, the GCF(35, 40, 100) is 5, as it is the largest number that divides all three numbers without leaving
a remainder.

Example 2. Find the GCF of 72 and 90.

Solution:
Let's list the factors of 72 and 90:

● Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

● Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Common factors: 1, 2, 3, 6, 9, 18

GCF(72, 90) = 18

Example 3. Find the GCF of 72 and 90.

Solution:

To find the Greatest Common Factor (GCF) of 45 and 80 using the division method (also known as the
Euclidean Algorithm), follow these steps:

Step 1: Divide the Larger Number by the Smaller Number

\[ \begin{align*}
80 \div 45 & = 1 \text{ with a remainder of } 35 \\
\end{align*} \]

Step 2: Replace the Larger Number

Replace the larger number (80) with the smaller number (45), and replace the smaller number (45) with the
remainder (35).

\[ \begin{align*}
45 \div 35 & = 1 \text{ with a remainder of } 10 \\
\end{align*} \]

Step 3: Continue the Process

Continue the process until the remainder is zero.

\[ \begin{align*}
35 \div 10 & = 3 \text{ with a remainder of } 5 \\
10 \div 5 & = 2 \text{ with a remainder of } 0 \\
\end{align*} \]

Step 4: Identify the GCF

The last non-zero remainder is 5. Therefore, the GCF(45, 80) = 5.

So, using the division method, the GCF of 45 and 80 is 5.

Example 4. Find the GCF of 184, 256, and 388 by the prime factorization method.

Solution:

To find the Greatest Common Factor (GCF) of 184, 256, and 388 using the prime factorization method, follow
these steps:

Step 1: Find the Prime Factorization of Each Number

● Prime factorization of 184: \[ 184 = 2^3 \times 23 \]

● Prime factorization of 256: \[ 256 = 2^8 \]
● Prime factorization of 388: \[ 388 = 2^2 \times 97 \]

Step 2: Identify the common prime factors among the prime factorizations:

Common factors: \(2^2 = 4\)

Step 3: Multiply the Common Prime Factors

Multiply the common prime factors to find the GCF:

\[ GCF(184, 256, 388) = 4 \]

Therefore, the GCF of 184, 256, and 388 is 4.

Example 5. What is the GCF of 300, 45, and 90?

To find the Greatest Common Factor (GCF) of 300, 45, and 90, you can use the prime factorization method.
Let's break down the prime factorization of each number:

● Prime factorization of 300: \[ 300 = 2^2 \times 3 \times 5^2 \]

● Prime factorization of 45: \[ 45 = 3^2 \times 5 \]
● Prime factorization of 90: \[ 90 = 2 \times 3^2 \times 5 \]

Now, identify the common prime factors:

Common factors: \( 3 \times 5 = 15 \)

Therefore, the GCF(300, 45, 90) is 15.

Practice Problems

Q1. What is the GCF of 17, 19 and 29?

a. 7
b. 9
c. 10
d. 1
Answer: d

Q2. Find the Greatest Common Factor (GCF) of 84, 72, and 90.
a. 8
b. 4
c. 6
d. 3
Answer: c

Q3. Find the Greatest Common Factor (GCF) of 1260 and 945.
a. 8
b. 310
c. 315
d. 6
Answer: c

Q4. Find the Greatest Common Factor (GCF) of 324, 468, and 540.
a. 6
b. 36
c. 40
d. 27
Answer: b

Q5. What is the GCF of 88, 56, and 120?

 a. 8
b. 12
c. 1
d. 4
Answer: a

Frequently Asked Questions

Here are some frequently asked questions about the Greatest Common Factor (GCF):

Q1. How is the GCF denoted?

Answer: The GCF is often denoted as GCF(a, b), where "a" and "b" are the numbers for which we want to find
the greatest common factor.

2. How do you find the GCF of two numbers?

Answer: The GCF can be found by listing the factors of each number identifying the common factors and
picking the smallest amongst the common factors. Another method involves using prime factorization.

3. Can the GCF be larger than the smallest number?

Answer: No, the GCF cannot be larger than the smallest number. It is always less than or equal to the
smallest number.

4. Can the GCF be negative?

Answer: No, the GCF is defined as a positive integer. It represents the largest positive number that divides the
given numbers without leaving a remainder.

5. Is the GCF unique for any set of numbers?

Answer: Yes, the GCF is unique for any set of numbers. However, if all numbers are zero, the GCF is
undefined.

6. How is the GCF used in simplifying fractions?

Answer: The GCF is used to simplify fractions by dividing both the numerator and denominator by their GCF.
This results in an equivalent fraction in its simplest form.

7. What is the GCF of prime numbers?

Answer: The GCF of two or more prime numbers is always 1 because prime numbers have no common
factors other than 1.