Greatest Common Factor

• 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 `45` and `80`.

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.

`80 \div 45 = 1` with a remainder of `35`

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)`.

`45 \div 35 = 1` with a remainder of `10`

Step `3`: Continue the process.

Continue the process until the remainder is zero.

`35 \div 10  = 3` with a remainder of `5`
`10 \div 5  = 2` with a remainder of `0`

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

Q`1`. What is the GCF of `17`, `19` and `29`?

1. `7`
2. `9`
3. `10`
4. `1`

Answer: d

Q`2`. Find the Greatest Common Factor (GCF) of `84`, `72`, and `90`.

1. `8`
2. `4`
3. `6`
4. `3`

Answer: c

Q`3`. Find the Greatest Common Factor (GCF) of `1260` and `945`.

1. `8`
2. `310`
3. `315`
4. `6`

Answer: c

Q`4`. Find the Greatest Common Factor (GCF) of `324`, `468`, and `540`.

1. `6`
2. `36`
3. `40`
4. `27`

Answer: b

Q`5`. What is the GCF of `88`, `56`, and `120`?

1. `8`
2. `12`
3. `1`
4. `4`

Answer: a

Frequently Asked Questions

Q`1`. 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.

Q`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.

Q`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.

Q`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.

Q`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.

Q`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.

Q`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`.