Common Factors: Definition, GCF, Examples

Common Factors

Have you ever thought about what numbers can clearly divide multiple other numbers? This section explores the concept of common factors, which is a foundational element in arithmetic. In simple terms, the Factors are numbers that divide a given number without leaving any remainder. When we identify factors that are shared by two or more numbers, these are known as common factors. This topic is important to simplify fractions, understand number relationships, and build skills for complex mathematical ideas.

When you break the concept into simple steps and clear examples,  the understanding of common factors becomes easier.  This section provides an in-depth explanation of how factors work, how to identify shared divisors, and why these concepts are essential in arithmetic and number theory.

Read more: What are the Factors of 36?

What Are Common Factors?

A factor represents an exact divisor of a particular number. This means that when a number gets divided by its factor, the remainder is always zero. Factors of any number will always be less than or equal to the number itself.  One (1) stands as a common factor for all numbers. Also, every number is a factor of itself. 

Sometimes you will notice certain factors appear in all lists when comparing the factors of different numbers. These shared factors are termed common factors.

For example, consider the factors of 35, which are {1, 5, 7, 35}. The factors for 45 are {1, 3, 5, 9, 15, 45}. Here, both 1 and 5 are present in both lists. Therefore, the common factors of 35 and 45 are 1 and 5. As per the formal definition, when two or more numbers are exactly divisible by the same number or numbers, those shared divisors become their common factors. It is a number that precisely divides a pair, or more, of numbers without any remainder.

Enroll Now for Mental Maths Online Classes

How to Find Common Factors?

Discovering the common factors between numbers is a straightforward process. It involves systematically listing the factors for each number. After that, spot those who appear in every list. This method ensures you capture all shared divisors. To find common factors, follow these steps:

Step 1: Write down all the individual factors for each number you are considering. It is often helpful to list these in separate rows.

Step 2: Examine these lists and identify the factors that are identical across all the numbers. These shared factors are your common factors.

For example, let's determine the common factors of 8, 12, 20, and 28.

• Factors of 8: 1, 2, 4, 8

• Factors of 12: 1, 2, 3, 4, 6, 12

• Factors of 20: 1, 2, 4, 5, 10, 20

• Factors of 28: 1, 2, 4, 7, 14, 28

By comparing these lists, you can see that 1, 2, and 4 are present in all four sets of factors. Thus, the common factors of 8, 12, 20, and 28 are 1, 2, and 4.

Read More: Factors of 60

Greatest Common Factor (GCF)

Beyond just identifying all common factors, we need to pinpoint the largest among them. This leads us to the Greatest Common Factor, GCF. The GCF signifies the largest number that can exactly divide two or more numbers without leaving a remainder. It is also known by other names such as the Highest Common Factor (HCF) or the Greatest Common Divisor (GCD). Once you have listed all the factors for a set of numbers and identified their common factors, the Greatest Common Factor (GCF) is simply the biggest number in that list of commonalities. 

If p and q are natural numbers, then GCF of p and q is the largest possible number capable of dividing both p and q precisely. Consider two numbers, 68 and 88, to illustrate finding their GCF:

• Factors of 68: 1, 2, 4, 17, 34, 68

• Factors of 88: 1, 2, 4, 8, 11, 22, 44, 88

The common factors shared by 68 and 88 are 1, 2, and 4. Among these, 4 is the largest. Therefore, the GCF of 68 and 88 is 4, which we can write as GCF(68, 88) = 4. This highest common factor is an essential concept for working with fractions and other number theory applications.

PW CuriousJr App

Key Rules for Common Factors

Rule 1: Finding Common Factors

To systematically determine the common factors for any given set of numbers, a two-step approach works reliably. This method simplifies the process of comparing multiple lists of divisors.

List All Factors: Begin by enumerating every factor for each individual number in question. A factor is any number that divides the original number completely, leaving no remainder.

Identify Shared Factors: Once you have complete lists for all numbers, scan through them. Any factor that appears in every single list is a common factor. Collect these shared numbers to form the complete set of common factors.

Rule 2: Identifying the Greatest Common Factor

The greatest common factor (GCF) is the pinnacle among the shared divisors. It represents the largest numerical value that divides all specified numbers exactly.

Determine All Common Factors: First, apply Rule 1 to find all the common factors of the numbers.

Select the Largest: From the set of common factors you've identified, simply choose the one with the highest value. This number is the GCF. This largest common factor is crucial in many mathematical contexts.

Also Read: HCF and LCM Formula

Make Maths Fun and Easy for Your Child with CuriousJr

Is your child struggling with maths at school or taking longer to complete homework? Many children need extra guidance and practice to fully understand concepts and gain confidence in solving problems. CuriousJr’s Online Maths Classes make learning simple, engaging, and enjoyable.

What makes CuriousJr online classes stands out:

• Live interactive sessions with easy-to-follow explanations

• Two-teacher model ensuring personal attention and instant doubt-solving

• Curriculum aligned with CBSE, ICSE, and other state boards

• Homework support and extra practice to strengthen key concepts

• Progress tracking and regular updates to parents, keeping you informed

Book a demo today and help your child build speed, accuracy, and confidence in maths.