VOOZH about

URL: https://www.geeksforgeeks.org/maths/factors-of-a-number/

⇱ Factors of a Number - GeeksforGeeks

  • Courses
  • Tutorials
  • Interview Prep

Factors of a Number

Last Updated : 23 Feb, 2026

A factor is a number that divides another number exactly, without leaving any remainder. Factors can also be seen as pairs of numbers that, when multiplied together, result in the original number.

Properties

  • Zero cannot be a factor of any number.
  • 1 is a factor of every number.
  • Every number greater than 1 has at least two factors: 1 and the number itself.
  • In elementary mathematics, we usually consider positive whole-number factors only.
  • A positive factor of a number is always less than or equal to the number.
  • 1 is the smallest positive factor, and the number itself is the greatest positive factor.
  • 2 is a factor of every even number.
  • 5 is a factor of every number ending in 0 or 5.

Finding Factors of a Number

We can find all the factors of a given number using three ways:

1. Multiplication Method

In this method, we find pairs of numbers that multiply together to give the original number.

Example: Find all the factors of 24 using the multiplication method.

Solution: We have to find all the pairs of whole numbers whose product is 12, like

👁 finding_factors_using_multiplication_method

Here the product of the following pairs is 24.

(1, 24) , (2, 12), (3, 8) and (4, 6)

Hence, all these numbers 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.

2. Division Method

In this method, we have to find all the divisors of the given number which are exactly divisible by it. Here we start dividing the given number by 1 and continue dividing it by the next number until we reach the number itself.

Example: Find all the factors of 12 using the division method.

Solution: We will take every natural number less than 12 and will check whether it is divisible by 12 or not

  • 12 ÷ 1 = 12 (remainder = 0)
  • 12 ÷ 2 = 6 (remainder = 0)
  • 12 ÷ 3 = 4 (remainder = 0)
  • 12 ÷ 4 = 3 (remainder = 0)
  • 12 ÷ 5 = 2 (remainder = 2)
  • 12 ÷ 6 = 2 (remainder = 0)
  • 12 ÷ 7 = 1 (remainder = 5)
  • 12 ÷ 8 = 1 (remainder = 4)
  • 12 ÷ 9 = 1 (remainder = 3)
  • 12 ÷ 10 = 1 (remainder = 2)
  • 12 ÷ 11 = 1 (remainder = 1)
  • 12 ÷ 12 = 1 (remainder = 0)

So, the numbers that are exactly divides 12 are 1, 2, 3, 4, 6, and 12. Hence these numbers are the factors of 12.

3. Factor Tree Method

A factor tree is a diagrammatic representation of the prime factors of a number. In this method, we find the factors of a number and then further factorize them until we get all the factors as prime numbers. Here, we consider the given number as the top of a tree and all its factors as its branches.

To find the prime factorization by factor tree method, we follow the below given steps:

  • First, split the given number (which is placed at the top of the tree) into factors.
  • Then write down the factor pair as the branches of the tree.
  • Again split the composite factors obtained in step 2.
  • Repeat steps 2 and 3 until all the factors become prime numbers.
  • Lastly, multiply all the prime factors obtained.

Example: Find the prime factors of 80 using the factor tree method.

Solution: We will break the number 80 into smaller factors and continue breaking them until we get only prime numbers.

👁 factor_tree_of_80

Now all the numbers obtained are prime numbers.

So, the prime factors of 80 are: 2, 2, 2, 2, and 5 = 2 ⁴ × 5

Factors Formulas

Let us suppose N is a natural number with prime factors X p × Y q × Z r, where

  • X, Y, Z are prime numbers and
  • p, q, r are their respective powers.

Name

Formula

Sum of Factors

[(X p+1-1)/X-1] × [(Y q+1-1)/Y-1] × [(Z r+1-1)/Z-1]

Numbers of Factors

(p+1) (q+1) (r+1)

Product of Factors

NTotal No. of Factors/2

Factors vs Multiples

Factors are the numbers which divides a number while multiples are the numbers which are obtained by multiplying a number with other number.

Factors

Multiples

Factors are the divisors of a number that divides the number without leaving any remainderMultiples are the product obtained by multiplying the number with other number
Every Number is a Factor of itselfEvery Number is a Multiple of itself
A number is the largest factor of itselfA number is the smallest multiple of itself
Number of Factors of a Number is finiteNumber of Multiples of a Number is infinite
Factor of a number is smaller or equal to the numberMultiple is equal or larger than the given number
Factor is found by dividing the numberMultiple is found by multiplication

Solved Examples

Example 1: Find all the factors of 64 by multiplication method.

Solution:

Factors of 64

  • 1 × 64 = 64
  • 2 × 32 = 64
  • 4 × 16 = 64
  • 8 × 8 = 64

Hence the factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

Example 2: Find common factors of 24 and 48.

Solution:

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48

Hence the common factors of 24 and 48 are 1, 2, 3, 4, 6, 8, 12, and 24.

Example 3: Find all the factors of 32 by division method.

Solution:

Factors of 32

  • 32 ÷ 1 = 32
  • 32 ÷ 2 = 16
  • 32 ÷ 4 = 8
  • 32 ÷ 8 = 4
  • 32 ÷ 16 = 2
  • 32 ÷ 32 = 1

Hence the factors of 32 are 1, 2, 4, 8, 16, and 32.

Example 4: Check if 50 is a factor of 1550 or not.

Solution:

We have to check the divisibility of 50 and 1550

1550 ÷ 50 = 31 ( with remainder 0)

Hence 1550 is exactly divisible by 50 so 50 is a factor of 1550.

Example 5: Check if 21 is a factor of 525 or not.

Solution:

We will check the divisibility of 525 with 21

On evaluating we get, 525 ÷ 21 = 25, with remainder 0

Here 525 is exactly divisible by 21. Hence 21 is a factor of 525.

Practice Questions on Factors of Number

Q1: Find all factors of 28 and 36.

Q2: Check if 12 is a factor of 144 or not.

Q3: Find prime factorization of 169.

Q4: Find prime factorization of 640.

Q5: Find common factors of 12, 24 and 48.

Comment

Explore