Factors of a Number: In mathematics, a factor is a number that divides another number perfectly, leaving no remainder. Factors can also be seen as pairs of numbers that, when multiplied together, result in the original number. For example, 2 and 3 are factors of 6 because multiplying them together gives 6. A single number can have multiple factors.

Let’s learn about the factors of prime and composite numbers, common factors, and methods to find them.

What are Factors of a Number?

Factors of a number can be defined as the divisors which divide the number exactly without leaving any number. Every number other than 1 has at least two factors, 1 and the number itself.

For example:

• Factors of 4 are 1, 2, and 4 as when we divide 1,2 and 4 by 4, it will evenly divide it by leaving 0 as the remainder.
• Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. As 24 is exactly divisible by all these numbers.

Factors of a Number Definition

In mathematics, factors of a number are defined as whole numbers that can be multiplied together to produce the original number. In other words, a factor of a number divides that number without leaving any remainder.

Factors of a Number Properties

These are some of the key properties of the factors of a number:

• Zero cannot be a factor of any number as division by 0 is not defined.
• 1 is a factor of every number.
• Every number has at least 2 factors, 1 and the number itself
• A factor can be negative.
• A factor of a number is always smaller than or equal to the number.
• 1 is the smallest factor of a number and the number itself is the greatest factor of the number.
• A factor of a number can never be a decimal or a fraction.
• 2 is a factor of every even number.
• 5 is a factor of every number that has 5 or 0 in its unit place.

How to Find Factors of a Number?

We can find all the factors of a given number in the following two ways:

• Multiplication method
• Division Method
• Factor Tree Method

Finding Factors Using Multiplication Method

In this method, we have to find all the pairs of whole numbers whose product is equal to the given number. Let us consider an example to understand that better.

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

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

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.

Finding Factors Using 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 square root of that number (or until we reach the number itself).

If the number exactly divides the original number then it is a factor else not. Let us consider an example to understand that better.

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.

Prime Factorization by 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.

How To Check whether a Number is a Factor?

To check whether the given number (first number) is a factor of another number (second number) or not we divide the numbers, if the remainder is 0 then it is a factor else not.

Example 1: 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.

Factors of Prime Numbers

Prime Numbers are the numbers that have exactly and only two factors, i.e. 1 and the number itself. For example: 2, 3, 19, 89, etc are prime numbers. To find all the factors of a prime number simply write 1 and the prime number itself.

• 2 is the only even prime number.
• The smallest prime number is 2.
• The smallest odd prime number is 3.

Factors of Composite Numbers

Composite Numbers are those that have more than two factors. For example: 4, 9, 25, 105, etc are composite numbers. All the other natural numbers (except 1) which are not prime are composite. 4 is the smallest composite number.

We can find all the factors of the composite number by using multiplication or division method.

Note: 1 has only one factor so it is neither prime nor composite.

Factors of Square Numbers

A square number can be defined as a product of some integer with itself. For example, 9 is a square number which is obtained by multiplying 3 by 3. Similarly, 16 is the square number obtained by multiplying 4 by 4. Factors of a square number can be obtained by using any method of factorization be it division method or multiplication method. For example,

• Factors of 16 are 1, 2, 4, 8, and 16.
• Similarly, factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

• A square number is always positive.
• The unit digit of a Square number can never be 2, 3, 7, or 8.
• The factors of square numbers 0 and 1 are 0 and 1 respectively.

Common Factors

The common factors are those factors that are common (same) to two or more numbers. These are the factors that divide each of them evenly.

Example: Find common factors of 10 and 15.

Solution:

Factors of 10 are: 1, 2, 5, 10

Factors of 15 are: 1, 3, 5, 15

We can see 1 and 5 are common in both 10 and 15. Hence 1 and 5 are common factors of 10 and 15.

Prime Factors of a Number

The prime numbers whose product gives back the original number are said to be the prime factors of that number. Prime factors can not be further divided.

For example 2 × 2 × 5 = 20, Hence the prime factors of 20 are 2, 2, and 5.

Factors Formulas

We can easily write any natural number as a product of its prime by using the prime factorization method discussed above.

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.

The basic factor formulas are:

• Sum of Factors
• Number of Factors
• Product of Factors