# Factors

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 1 What are Factors? 2 Properties of Factors 3 How to Find Factors? 4 Tips and Tricks 5 Factors Calculator 6 Factoring in Math 7 Finding Number of Factors 8 Types of Factoring 9 Solved Examples on Factors 10 Challenging Questions on Factors 11 Practice Questions on Factors 12 Maths Olympiad Sample Papers 13 Frequently Asked Questions (FAQs) 14 FREE Worksheets on Factors

## What are Factors?

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

For example,

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

Are there any factors of $$12$$ apart from $$2$$ and $$6$$?

Yes, $$1, 3, 4$$ and $$12$$ also divide $$12$$ along with $$2$$ and $$6$$

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.

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
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 $$25$$

## 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 $$108$$

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:

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.

 False
 Example 3

Find the number of factors of $$1620$$

Solution:

Find the prime factorization of $$1620$$

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 $$\mathbf{30}$$

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.

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.

## 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 polynomials
• Factoring by grouping

## 3. What is full factoring?

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

Divisibility
Divisibility
Factors and Multiples
Factors and Multiples
Factors, Multiples, and Primes
Factors, Multiples, and Primes
Divisibility