Common Factors: Definition, Properties and Examples

Common Factors: Let us first define a factor before we can understand the concept of common factors. When a number is multiplied by another number to produce another number, the multiplied number is referred to as a factor. So, the factors which are common to two or more numbers are known as common factors.

This article will cover the study of common factors and will expertise ourselves in finding the common factors and the highest common factor.

Common Factors

Common factors can be defined as the factors that are common to two or more numbers. Thus, a common factor is a number with which a set of two or more numbers will be divided exactly.

To find common factors of two numbers, first, list out all the factors of two numbers separately and then compare them. Now, write the factors which are common to both the numbers. These factors are called common factors of given two numbers. 

How to Find Common Factors?

The factors are the numbers that divide the original number completely. Therefore, to find the common factors, we need to follow these steps:

Step 1: Take the two numbers separately.
Step 2: Write the factors of each number separately.
Step 3: Now compare and find which numbers are common in both factors.
Step 4: Those are the common factors for the given two numbers.
Step 5: The same procedure can be followed when finding common factors for more than two numbers as well.

Common Factor Examples

To understand the concept of finding common factors more easily and clearly. Let us learn to find the common factors with the help of a couple of examples.

Example 1: Find the common factors of \(15\) and \(20\).
Let us check the factors of the two numbers, i.e., \(15\) and \(20\).
Factors of \(15\) are \(1, 3, 5, 15.\)
Factors of \(25\) are \(1, 5, 10, 20\)
We can see that \(1\) and \(5\) are present as factors in both the numbers.
Hence, \(1, 5\) are the common factors of \(15\) and \(20\).

Example 2: Find the common factors of \(6\) and \(12\).
Let us check the factors of the two numbers, i.e., \(6\) and \(12\).
Factors of \(6\) are \(1, 2, 3, 6.\)
Factors of \(12\) are \(1, 2, 3, 4, 6, 12.\)
We can see, both \(6\) and \(12\) have \(1, 2, 3, 6\) as the common factors.
Hence, \(1, 2, 3, 6\) are the common factors of \(6\) and \(12\).

Example 3: Find the common factors of \(12, 24\) and \(48\).
Let us check the factors of the numbers, i.e., \(12, 24\) and \(48\).
Factors of \(12\) are \(1, 2, 3, 4, 6, 12.\)
Factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12, 24.\)
Factors of \(48\) are \(1, 2, 3, 4, 6, 12, 16, 24, 48\).
We can see, \(12, 24\) and \(48\) have \(1, 2, 3, 4, 6, 12\) as the common factors.
Hence, \(1, 2, 3, 4, 6, 12\) are the common factors of \(12, 24\) and \(48\).

In the same way, we can find the common factors for n numbers in a set. In this method, we can also find the highest common factor. Among all the obtained common factors, pick the greatest common factor and that factor is termed as the greatest common factor.

Highest Common Factor (HCF)

HCF stands for Highest Common Factor, and the HCF of two or more given numbers is the greatest number that completely divides each of the given numbers.
Let us understand the concept of HCF with the help of examples.
1. The greatest number that can divide both \(18\) and \(24\) completely is \(6\), and therefore, the HCF of \(18\) and \(24\) is \(6\).
2. HCF of numbers \(5, 10\) and \(15\) is \(5\). This is because \(5\) is the greatest number that divides each of the given numbers \(5, 10\) and \(15\) completely.

Methods of Finding Highest Common Factor

For finding the HCF of two or more given numbers, we have
1. Common Factor Method
2. Prime Factor Method
3. Division Method

Let us discuss and understand each method one by one.

Common Factor Method

The following steps need to be followed to find the HCF by the common factor method. They are:

Step 1: Find all the possible factors of each given number.
Step 2: From the factors obtained in the previous step, select the common factors.
Step 3: Out of the common factors obtained in the previous step, take the highest factor, which is the highest common factor (HCF) of the given numbers.

Example: Find the common factors of \(36\) and \(48\).
Factors of \(36= 1, 2, 3, 4, 6, 9, 12,\) \(18\) and \(36\).
Factors of \(48= 1, 2, 3, 4, 6, 8, 12, 16,\) \(24\) and \(48\).
Factors common in \(36\) and \(48= 1, 2, 3, 4, 6\) and \(12\).
The highest among the common factors is \(12\).

Hence, the highest common factor (HCF) of \(36\) and \(48\) is \(12\).

Prime Factor Method

The following steps need to be followed to find the HCF by the prime factor method. They are:

Step 1: Split each given number into its prime factors.
Step 2: Highlight the common prime factors.
Step 3: Multiply the prime factors obtained in the previous step.
The product so obtained is the HCF of the given numbers.

Example: Find the HCF of \(18, 24\) and \(60\).
Prime factorisation of \(18= 2×3×3\)
Prime factorisation of \(24= 2×2×2×3\)
Prime factorisation of \(60= 2×2×3×5\)
Common prime factors are \(2\) and \(3\).
HCF = Product of common factors \(= 2×3=6.\)

Hence, the HCF of \(18, 24\) and \(60 = 6\).

Division Method

Follow the steps to find the HCF by division method. They are:

Step 1: Divide the greater number by the smaller number.
Step 2: By the remainder obtained in step \(1\), divide the smaller number.
Step 3: By the remainder obtained in step \(2\), divide the remainder obtained in step \(1\).
Step 4: Continue the process in the same way till no remainder is left.

Let us find the HCF of \(275\) and \(525\) by the long division method.

Step 1: Write \(275\) as the divisor and \(525\) as the dividend and divide.

Step 2: The remainder becomes the new divisor. The last divisor, \(275\), is the new dividend.
\(275 ÷ 250\) gives \(1\) as the quotient and \(25\) as the remainder.

Step 3: Now \(25\) becomes the new divisor, and \(250\) the new dividend. 
\(250 ÷ 25\) gives \(10\) as the quotient and \(0\) as the remainder. 
So, the last divisor \(25\) is the HCF.

Solved Examples – Common Factors

Q.1. Find the HCF of \(15, 20\) and \(25\).
Ans: First, find the factors of all the given numbers.
Factors of \(15\) are \(1, 3, 5, 15\)
Factors of \(20\) are \(1, 2, 4, 5, 10, 20\)
Factors of \(25\) are \(1, 5, 25\)
Common factors are \(1, 5.\)
Thus, the highest common factor \(= 5\)

Q.2. Find the HCF of \(24, 12, 36\) and \(60\) with the help of prime factorisation.
Ans: The prime factorising each number, we get
\(24 = 2 \times 2 \times 2 \times 3\) 
\(12 = 2×2×3\)
\(36= 2×2×3×3\)
\(60 = 2×2×3×5\)
The prime factors common to the given numbers are \(2, 2\) and \(3\).
Therefore, HCF \(= 2×2×3 = 12.\)

Q.3. Find the HCF of \(36\) and \(60\) by the division method.
Ans: Follow the steps to find the HCF by division method. 
a. Divide the greater number by the smaller number.
b. By the remainder obtained in the previous step, divide the smaller number.
c. By the remainder obtained in the previous step, divide the remainder obtained in the first step.
d. Continue the process in the same way till no remainder is left.

Since the last divisor is \(12,\) therefore the HCF is \(12\).

Q.4. Using the common factor method, find the HCF of \(16, 32\) and \(49\).
Ans: Let us find the factors of \(16, 32\) and \(49\).
Factors of \(16\) are \(1, 2, 4, 8, 16\).
Factors of \(32\) are \(1, 2, 4, 8, 16, 32.\)
Factors of \(49\) are \(1, 7, 49.\) 
Common factor \(= 1\)
Therefore, the required HCF \(= 1\)

Q.5. Find the common factor of \(6, 12, 24\) and \(48\).
Ans: Let us find the factors of the numbers, i.e., \(6, 12, 24\) and \(48\).
Factors of \(6\) are \(1, 2, 3, 6.\)
Factors of \(12\) are \(1, 2, 3, 4, 6, 12.\)
Factors of  \(24\) are \(1, 2, 3, 4, 6, 8, 12, 24\).
Factors of \(48\) are \(1, 2, 3, 4, 6, 12, 16, 24, 48\).
We can see, \(6, 12, 24\) and \(48\) have \(1, 2, 3, 6\) as the common factor.

Summary

In this article, we learned about the definition of common factors with the help of examples. We also understood the concept of HCF and how finding the common factors helps in finding the highest common factor. Finally, we mastered ourselves by learning the various methods of finding the HCF with the help of examples. 

Frequently Asked Questions (FAQ) – Common Factors

Q.1. What are the common factors of \(3\) and \(5\) ?
Ans: Since \(3\) and \(5\) are twin prime numbers, and twin prime numbers do not have any other common factors except \(1\), the only common factor between \(3\) and \(5\) is \(1\).

Q.2. What are common factors?
Ans:A common factor is a number with which a set of two or more numbers will be divided exactly.

Q.3. What are the common factors of \(12\) and \(15\)?
Ans: Let us check the factors of the two numbers, i.e., \(12\) and \(15\).
Factors of \(12\) are \(1, 2, 3, 6, 12\).
Factors of \(15\) are \(1, 3, 5, 15\).
We can see, both \(12\) and \(15\) have \(1, 3\) as the common factor.

Q.4. How do you find common factors?
Ans: Therefore, to find the common factors, we need to follow these steps.
(a). Take the two numbers separately.
(b). Write the factors of each given number.
(c). Now compare and find which numbers are common in both factors.
(d). Those are the common factors for the given two numbers.

Q.5. What are the common factors of \(10\) and \(15\)?
Ans: Let us check the factors of the two numbers, i.e., \(10\) and \(15\).
Factors of \(10\) are \(1, 2 , 5, 10\).
Factors of \(15\) are \(1, 3, 5, 10.\)
We can see, both \(10\) and \(15\) have \(1, 5\) as the common factor.