Divisors in Maths (original) (raw)

Last Updated : 23 Jul, 2025

**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.

Table of Content

• Divisor Definition
• How to Find Divisors of a Number?
• Properties of Divisors
• Divisors and Dividends
• Divisors vs Factors
• Divisors Examples
• Sample Problems on Divisors
• Practice Problems on Divisors in Maths

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) . . . (xk + 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) . . . (xk + 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/p 1 ) × (1 - 1/p 2 ) × . . . × (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.