Factors

What are Factors?

A factor is a number that divides the given number without any remainder.

For example,

The factors of 12 are 2 and 6. The product of 2 and 6 gives 12.

\(2\) and \(6\) are some of the factors of \(12\) as each of them divides \(12\) without any remainder.

What are Factors? - 2 and 6 are the factors of 12. When you divide 12 by 2, the remainder is 0.What are Factors? - 2 and 6 are the factors of 12. When you divide 12 by 6, the remainder is 0.

Are there any factors of \(12\) apart from \(2\) and \(6\)?

Yes, \(1, 3, 4\) and \(12\) also divide \(12\) along with \(2\) and \(6\)

What are Factors? - When you divide 12 by 1, the remainder is 0.What are Factors? - When you divide 12 by 3, the remainder is 0.

What are Factors? - When you divide 12 by 4, the remainder is 0.What are Factors? - When you divide 12 by 12, the remainder is 0.

Thus, the factors of \(12\) are \(1,2,3,4,6\) and \(12\)

Apart from these, there can be negative factors of \(12\) as well such as \(-1,-2,-3,-4,-6\) and \(-12\)


Properties of Factors

The properties of factors are:

  • The number of factors of a number is finite.
  • A factor of a number is always less than or equal to the given number.
  • Every number except \(0\) and \(1\) has at least two factors, \(1\) and itself.
  • Division is the operation that is used in finding the factors.

How to Find Factors?

To find the factors of a number, we need to do the following:

  • Find all the numbers less than or equal to the given number.
  • Divide the given number by each of the numbers.
  • The divisors that give the remainder to be \(0\) are the factors of the number.

Example:

Find the positive factors of \(6\)

Solution:

The positive numbers that are less than or equal to \(6\) are \(1,2,3,4,5\) and \(6\)

Let us divide \(6\) by each of these numbers.

How to find factors? - When you divide 6 by 1, the remainder is 0.How to find factors? When you divide 6 by 2, the remainder is 0.How to find factors? - When you divide 6 by 3, the remainder is 0.

How to find factors? - When you divide 6 by 4, the remainder is 2.How to find factors? - When you divide 6 by 5, the remainder is 1.How to find factors? - When you divide 6 by 6, the remainder is 0.

We can observe that the divisors \(1,2,3\) and \(6\) give zero as the remainder.

Hence,

Factors of 6 are \(\mathbf{1,2,3 \text{ and } 6}\)
 
tips and tricks
Tips and Tricks
  1. To find the factors, we can apply the divisibility rules.
  2. The numbers greater than half of a given number (except the number of itself) CANNOT be the factors of the given number.
    For example, while finding the factors of \(50\), it is sufficient to check for the numbers from \(1\) to \(25\)
    Tips and tricks to find factors. To find the factors of 50, it is sufficient to check the numbers from 1 to 25 and the number 50.

Factors Calculator

Here is the "Factors Calculator".

We can enter any number here and it will show the factors of the given number with a step-by-step explanation.


Factoring in Math

The meaning of factoring in Math is representing a number as the product of two or more given numbers.

The numbers of this product will be the factors of the given number.

Examples:

  • \(12 = 6 \times 2\)
  • \(35 = 7 \times 5\)
  • \(108= 12 \times 9\)
  • \(27 = 9 \times 3\)

Finding Number of Factors

We can find the number of factors of a given number using the following steps.

  • Find its prime factorization i.e., express it as the product of primes.
  • Write the prime factorization in the exponent form.
  • Add \(1\) to each of the exponents.
  • Multiply all the resultant numbers.
  • This product would give the number of factors of the given number.

Example:

Find the number of factors of the number \(108\)

Solution:

Find the prime factorization of the number \(108\)

Finding Number of Factors: Prime factorisation of 108 is shown.

Thus,

\[108 = 2 \times 2 \times 3 \times 3 \times 3 \]

Write this prime factorization in the exponent form.

\[108 = 2^2 \times 3^3\]

Add \(1\) to each of the exponents, \(2 \text{ and } 3\), here.

Then:

\[2+1=3\\3+1=4\]

Multiply these numbers:

\[3 \times 4 = 12\]

Thus, 

Number of factors of \(\mathbf{108}\) is \(\mathbf{12}\)

The actual factors of \(108\) are \(1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54\) and \(108\)

Here, \(108\) has \(12\) factors and hence our above answer is correct.


Types of Factoring

There are various types of factoring in Algebra.

Some of them are as follows:

We will learn about these types of factoring in higher grades.

Click on the above links to learn each of them in detail.


Solved Examples

Example 1

 

 

Find the positive factors of \(64\)

Solution:

Half of the given number is \(32\)

To find the factors of \(64\), it is sufficient to check the numbers from \(1\) to \(32\) whether they give the remainder zero when we divide \(64\) by them.

We can observe that the numbers \(1,2,4,8,16\) and \(32\) give the remainder \(0\) when we divide \(64\) by each of them.

Thus, the factors of \(\mathbf{64}\) are

\(\mathbf{1,2,4,8,16}\)  and  \(\mathbf{32}\)
Example 2

 

 

Which of the following statement(s) is/are true?

  1. The factor of a number can be greater than the number.
     
  2. Some numbers can have infinite number of factors.

Solution:

  1. The first statement, "The factor of a number can be greater than the number," is FALSE.

We know that factors are the divisors of the number that leave \(0\) as the remainder.

Hence, they are always less than the number.

False
  1. The second statement, "Some numbers can have infinite number of factors," is FALSE.

The number of factors of a number is finite.

Thus, the answers are:

False
Example 3

 

 

Find the number of factors of \(1620\)

Solution:

Find the prime factorization of \(1620\)

Finding Number of Factors: Prime factorisation of 1620 is shown.

Thus,

\[1620 = 2^{2} \times 3^{4} \times 5^{1}\]

Add \(1\) to each of the exponents.

We get:

\[2+1=3\\4+1=5\\1+1=2\]

Find the product of all these numbers:

\[3 \times 5 \times 2 = 30\]

Thus,

Number of factors of \(\mathbf{1620} \) is \(\mathbf{30}\)
 
Challenge your math skills
Challenging Questions
  1. Find the factors of \(124\)
  2. Find the number of factors of \(124\)

Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

 


Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How do you factorise in maths?

Factorising is the method of representing a number as the product of two or more given numbers.

The numbers of this product will be the factors of the given number.

Examples:

  • \(12 = 6 \times 2\)
  • \(35 = 7 \times 5\)
  • \(108= 12 \times 9\)
  • \(27 = 9 \times 3\)

2. What are the 6 types of factoring?

The meaning of factoring is representing a number as the product of two or more given numbers.

Some types of factoring in algebra are:

  • Factoring the HCF
  • Factoring using algebraic identities
  • Factoring expressions
  • Factoring quadratic expressions
  • Factoring polynomials
  • Factoring by grouping

3. What is full factoring?

Full factoring of a number means the "prime factorization" of the number.

 
Download Free Grade 4 Worksheets
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Divisibility
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