Factors

A factor is a Latin word, and it means "a doer" or "a maker" or "a performer." A factor of a number in math is a number that divides the given number. Hence, a factor is nothing but a divisor of the given number. To find the factors, we can use multiplication as well as division method. We can also apply the divisibility rules.

Factoring is a useful skill to find factors, which is further utilized, in real life situations, such as dividing something into equal parts or dividing in rows and columns, comparing prices, exchanging money and understanding time and making calculations, during travel.

Table of Contents

What Are Factors?

In math, a factor is a number that divides another number evenly, that is with no remainder. Factors can be algebraic expressions as well, dividing another expression evenly. Well, factors and multiples are a part of our daily life, from arranging things, such as sweets in a box, handling money, to finding patterns in numbers, solving ratios, and working with expanding or reducing fractions.

Factor Definition 

A factor is a number that divides the given number without any remainder. Factors of a number can be referred to as numbers or algebraic expressions that evenly divide a given number/expression. The factors of a number can either be positive or negative.

For example, let's check for the factors of 8. Since 8 can be factorized as 1 x 8 and 2 x 4 and we know that the product of two negative numbers is a positive number only. Therefore, the factors are 8 are actually 1, -1, 2, -2, 4, -4, 8 and -8. But when it comes to problems related to the factors, only positive numbers are considered, that too a whole number and a non-fractional number.

Properties of Factors

Factors of a number have a certain number of properties. Given below are the properties of factors:

  • 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 and multiplication are the operations that are used in finding the factors.

How to Find Factors of a Number?

We can use both "Division" and "Multiplication" to find the factors.

Factors by Division

To find the factors of a number using division:

  • 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

(by the definition of a factor of a number)

Example: Find the positive factors of 6 using division.

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.

Finding Factors by Division Method

We can observe that the divisors 1, 2, 3, and, 6 give zero as the remainder. Thus, factors of 6 are 1, 2, 3, and 6.

Factors by Multiplication

To find the factors using the multiplication:

  • Write the given number as the product of two numbers in different possible ways.

  • All the numbers that are involved in all these products are the factors of the given number.

Example: Find the positive factors of 24 using multiplication.

Solution:

We will write 24 as the product of two numbers in multiple ways.

Finding Factors by Multiplication

All the numbers that are involved in these products are the factors of the given number (by the definition of a factor of a number)

Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Finding the Number of Factors

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

  • Step 1: Find its prime factorization, i.e. express it as the product of primes.
  • Step 3: Write the prime factorization in the exponent form.
  • Step 3: Add 1 to each of the exponents.
  • Step 4: 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:

Perform prime factorization of the number 108:

Finding the Number of Factors

Thus, 108 = 2 x 2 x 3 x 3 x 3. In the exponent form: 108 = 22 x 33. Add 1 to each of the exponents, 2 and 3, here. Then, 2 + 1 = 3, 3 + 1 = 4. Multiply these numbers: 3 x 4 = 12. Thus, Number of factors of 108 is 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.

Algebra-Factors

Factors do exist for an algebraic expression as well. For example, the factors of 6x are 1, 2, 3, 6, x, 2x, 3x, and 6x. There are different types of procedures to find factors 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.


FAQs on Factors

How do you Factor Equations?

We cannot actually factor equations, but we can factor expressions. Factoring an expression is writing it as the product of two or more expressions. For example: 3x+ 6x = 3x(x + 2)

What is Prime Factorization?

The prime factorization of a number is writing it as the product of two or more prime numbers. For example: The prime factorization of 60 = 22 x 3 x 5.

How to Find 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.

What are the Common Factors of 4 and 12?

Factors of 4 = 1, 2 and 4. Factors of 12 = 1, 2, 3, 4 , 6 and 12. Thus, common factors of 4 and 12 are 1, 2 and 4.

What is a Factors Formula?

The factors formula for a number gives the total number of factors of a number. For a number N, whose prime factorization is Xa x Yb x Zc, (a+1) (b+1) (c+1) is the total number of factors.

What is a Factor of Every Number?

A factor of a number is a number that is totally divisible by that number. 1 divides every number, thus,1 is a factor of every number.

What are the Prime Factors of a Number?

The prime factor of a number is the factor of the given number which is a prime number. Factors are the numbers that multiply together to give another number.

Can 0 be a Factor of Any Number?

Since n/0 is undefined for any number, other than zero. Therefore, zero is not a factor of any non-zero number. Other than zero, all integers are factors of zero because 0n = 0, for all numbers.


Important Topics

Given below is the list of topics that are closely connected to Factors. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

Solved Examples on Factors

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. Therefore, the factors of 64 are 1, 2, 4, 8, 16, and 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 an infinite number of factors.

Solution: 

1. The 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. Therefore, the answer is: False

2. The statement, "Some numbers can have an infinite number of factors," is FALSE. The number of factors of a number is finite. Therefore, the answer is: False.

Example 3: Find the number of factors of 1620

Solution:

Find the prime factorization of 1620

Solved Example: Number of factors of 1620

Thus, 1620 = 22 x 34 x 51

Add 1 to each of the exponents. We get: 2 + 1 = 3, 4 + 1 = 5,1 + 1 = 2. The product of all these numbers: 3 x 5 x 2 = 30. Therefore, the number of factors of 1620 is 30.

Practice Questions on Factors

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

 
 
 
 
 
 
Download Free Grade 4 Worksheets
Divisibility
Grade4 | Worksheet 3
Divisibility
Grade4 | Worksheet 4
Factors and Multiples
Grade4 | Worksheet 1
Factors and Multiples
Grade4 | Worksheet 2
Download Free Grade 5 Worksheets
Factors, Multiples, and Primes
Grade5 | Worksheet 2
Factors, Multiples, and Primes
Grade5 | Worksheet 1
Divisibility
Grade5 | Worksheet 1
Divisibility
Grade5 | Worksheet 2
  
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