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?
• Properties of Factors
• How to Find the Factors of a Number?
• Finding the Number of Factors
• Algebra Factors
• FAQs on Fractions
• Solved Examples
• Practice Questions

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.

• 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.

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.

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.

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.

• 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:

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.

• Factoring Expressions
• Factoring Quadratic Expressions
• Factoring Polynomials

We will learn about these types of factoring in higher grades. Click on the above links to learn each of them in detail.

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: 3x² + 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 X^a x Y^b x Z^c, (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.

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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.

Factors of 2 Factors of 4 Factors of 5 Factors of 6 Factors of 7 Factors of 8 Factors of 9 Factors of 10 Factors of 11 Factors of 12 Factors of 13 Factors of 14 Factors of 15 Factors of 16 Factors of 17 Factors of 19 Factors of 20 Factors of 21 Factors of 22 Factors of 23 Factors of 24 Factors of 25 Factors of 26 Factors of 27 Factors of 28 Factors of 29 Factors of 30 Factors of 31 Factors of 32 Factors of 33 Factors of 34 Factors of 35 Factors of 37 Factors of 38 Factors of 39 Factors of 40

Factors of 41 Factors of 42 Factors of 43 Factors of 44 Factors of 46 Factors of 47 Factors of 48 Factors of 49 Factors of 50 Factors of 51 Factors of 52 Factors of 53 Factors of 54 Factors of 55 Factors of 56 Factors of 57 Factors of 58 Factors of 59 Factors of 61 Factors of 62 Factors of 63 Factors of 65 Factors of 66 Factors of 67 Factors of 68 Factors of 69 Factors of 70 Factors of 71 Factors of 72 Factors of 73 Factors of 74 Factors of 75 Factors of 76 Factors of 77 Factors of 78 Factors of 79 Factors of 80

Factors of 81 Factors of 84 Factors of 85 Factors of 86 Factors of 87 Factors of 88 Factors of 89 Factors of 90 Factors of 91 Factors of 92 Factors of 93 Factors of 96 Factors of 97 Factors of 98 Factors of 99 Factors of 104 Factors of 105 Factors of 108 Factors of 112 Factors of 117 Factors of 119 Factors of 121 Factors of 125 Factors of 128 Factors of 135 Factors of 140 Factors of 144 Factors of 147 Factors of 150 Factors of 160 Factors of 168 Factors of 175 Factors of 180 Factors of 192 Factors of 200

Factors of 216 Factors of 224 Factors of 243 Factors of 245 Factors of 250 Factors of 256 Factors of 270 Factors of 288 Factors of 294 Factors of 300 Factors of 320 Factors of 360 Factors of 400 Factors of 441 Factors of 450 Factors of 512 Factors of 625 Factors of 90

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

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.