Divisors in Maths

In Mathematics, Divisor is the number from which another number is divided, known as the dividend, to determine the quotient and remainder. In arithmetic, division is one of the four fundamental operations; other operations are addition, subtraction, and multiplication.

In this article, we will discuss both definitions of a divisor, including the general, and the definition in number theory.

Divisor Definition

Divisors are integers that are used to divide another number . For example, the 5 is dividing 16 then 5 will be the divisor , 16 willl be the dividend.

Divisors play a fundamental role that is frequently used in various areas of mathematics such as number theory, algebra, arithmetic, and problem-solving situations.

In other words, a divisor is an integer that, when multiplied by a whole number, results in the original number without any fractional or remainder part.

How to Find Divisors of a Number?

There are a few ways to find out divisors of a number:

Brute Force Method: List out all the factors of the number, beginning with 1 and itself. For example, the divisors of 20 are 1, 2, 4, 5, 10, and 20.

Divisor Using Prime Factorization: Prime Factorization of a number involves expressing a positive integer as a product of its prime divisors. Prime numbers are those that have only 2 factors i.e. 1 and themselves.

For example, 20 can be written as follows:

20 = 5 × 4

Here, 5 is a prime number and 4 is a composite number, which can have more factors.

Thus, 20 = 2 × 2 × 5

Hence, the prime factorization of 20 is 2 × 2 × 5.

Properties of Divisors

Here are some interesting properties of divisors:

• A divisor can never be 0 i.e., any number with 0 as its divisor is undefined. For example: If 2 is divided by 0 then the remainder will be 0 (2 ÷ 0 = not defined).
• A divisor can be positive or negative. However, if stated otherwise, the term is often restricted to positive integers.
• If a divisor has the same value as a dividend, the quotient will always be 1.
• An even number always has an even divisor. An odd number will always have an odd divisor.
• If a divisor is a decimal number, it should be converted into a whole number before dividing.

Divisors and Dividends

The key differences between both divisors and dividends are listed in the following table:

In other words, a divisor is a factor of a dividend. For instance:

For 12 ÷ 4 = 3

• The divisor in this expression is 4
• The dividend in this expression is 12
• The quotient/remainder in this expression is 3.

Read more: Dividend, Quotient and Remainder.

Divisors vs Factors

The key difference between divisors and factors are:

Let’s consider an example, 15 divided by 5 gives 3 (15 ÷ 5 = 3). Similarly, if we divide 15 by 3 it gives 5 (15 ÷ 3 = 5).

The factors and divisors of 15 are 1, 3, 5 and 15 respectively.

However, if we divide 15 by 2 it will not completely divide the number. Thus, 2 is not the factor of 15 but will be considered a divisor.

Divisors Examples

As we already discussed in number theory, divisors are those positive integers that can evenly divide into another natural number without leaving a remainder. In other words, if you have an integer "n," its divisors are the integers that, when multiplied by another integer, result in "n" without a remainder. Some examples of divisors for various natural numbers are discussed as below:

Divisors of 60

A number is a divisor of 60 if it divides 60 completely. Thus, x is a divisor of 60 if 60 divided by x is an integer. We have:

Thus all the divisors for 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. So, it is concluded that 60 has a total of twelve divisors.

Divisors of 72

A number is a divisor of 72 if it divides 72 completely. Thus, x is a divisor of 72 if 72 divided by x is an integer. We have:

Thus all the divisors for 60 are 1, 2, 3, 4, 6, 8, 9, 12, 24, 36 and 72. So, it is concluded that 60 has a total of eleven divisors.

Divisors of 18

A number is a divisor of 18 if it divides 18 completely. Thus, x is a divisor of 18 if 18 divided by x is an integer. We have

Thus all the divisors for 60 are 1, 2, 3, 6, 9 and 18. So, it is concluded that 18 has a total of six divisors.

Some other examples of divisors includes:

• The divisors of 12 are 1, 2, 3, 4, 6, and 12.
• The divisors of 20 are 1, 2, 4, 5, 10, and 20.
• The divisors of 7 are 1 and 7.
• The divisors of 16 are 1, 2, 4, 8, and 16.
• The divisors of 25 are 1, 5, and 25.
• The divisors of 9 are 1, 3, and 9.
• The divisors of 1 are 1.
• The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• The divisors of 33 are 1, 3, 11, and 33.
• The divisors of 50 are 1, 2, 5, 10, 25, and 50.

Read More,

• Addition
• Subtraction
• Multiplication
• Divisibility Rules

Sample Problems on Divisors

Problem 1: Find all the Divisors of 14.

Solution:

• 1 x 14 = 14
• 2 x 7 = 14
• 7 x 2 = 14

The divisors of the number 14 are 1, 2, 7 and 14

Therefore, the number of divisors of 14 is 4.

Problem 2: Find all the Divisors of 24

Solution:

• 1 x 24 = 24
• 2 x 12 = 24
• 3 x 8 = 24
• 4 x 6 = 24
• 6 x 4 = 24

The factors of the number 24 are 1, 2, 3, 4, 6 and 24

Therefore, the number of divisors of 24 is 6.

Problem 3: Find prime divisors of 460

Solution:

Step 1: Take the number 460 and and divide it with the smallest prime factor 2

• 460 ÷ 2 = 230

Step 2: Again divide 230 by the smallest prime factor 2 as 230 is divisible by 2

• 230 ÷ 2 = 115

Step 3: Now, 115 is not completely divisible by 2 as it will give a fractional number. Let’s divide it with the next smallest prime factor 5

• 115 ÷ 5 = 23

Step 4: As 23 is itself a prime number as it can only be divided by itself and 1. We cannot proceed further.

Therefore, the prime divisors of number 460 are 2² x 5 x 23

Problem 4: Find the number of divisors of 48.

Solution:

It is calculated using the following formula:

d(n) = (x₁ + 1) (x₂ + 1) . . . (x_k + 1)

The prime factorization of 48 is 2⁴ x 3. Therefore, the number of divisors of 12 are:

⇒ d(48) = (4 +1) (1+1)

⇒ d(48) = 5 x 2

⇒ d(48) = 10

Therefore, there are 10 divisors of 48

Problem 5: Find the number of divisors of 1080

Solution:

It is calculated using the following formula:

d(n) = (x₁ + 1) (x₂ + 1) . . . (x_k + 1)

The prime factorization of 1080 is 2³ x 3³ x 5. Therefore, the number of divisors of 1080 are:

⇒ d(1080) = (3+1) (3+1) (1+1)

⇒ d(1080) = 4 × 4 × 2

⇒ d(1080) = 32

Therefore, there are 32 divisors of 1080

Problem 6: Find the sum of divisors of 52

Solution:

It is calculated using the following formula:

σ(n) = 1 + n + ∑_{d|n, d≠1, d≠n} d

The divisors of 52 are 1, 2, 4, 13, 26, and 52

⇒ σ(52)=1+52+(2+4+13+26)

⇒ σ(52)=53+2+4+13+26

⇒ σ(52)=98

Therefore, the sum of divisors of 52 is 98

Problem 7: Find the φ(60) using Euler's Totient Function formula

Solution:

It is calculated using the following formula:

ϕ(n) = n × (1 - 1/p₁) × (1 - 1/p₂) × . . . × (1 - 1/p_k)

The prime factorization of 60 is 2² x 3 x 5

⇒ ϕ(60) = 60 x ( 1 - 1/2) x (1 - 1/3) x (1 - 1/5)

⇒ ϕ(60) = 60 (1/2) (2/3) (4/5)

⇒ ϕ(60) = 24

So, φ(60) = 24. There are 24 positive integers less than or equal to 60 that are relatively prime to 60.

Practice Problems on Divisors in Maths

Problem 1: List all divisors of following.

• 12
• 20
• 48
• 42
• 144

Problem 2: Find the sum of all the divisors of the number 28.

Problem 3: Determine the number of divisors of 36.

Problem 4: Calculate the sum of all the divisors of 120.

Problem 5: How many divisors does the number 150 have?

Problem 6: Find the sum of the divisors of 72.

Problem 7: Determine the number of divisors of 144.

Problem 8: Calculate the sum of the divisors of 210.

Problem 9: How many divisors does the number 625 have?

Problem 10: Find the sum of all the divisors of 90.

Problem 10: Calculate the product of all divisors of a number 6.