The Logic Behind Number of Factors – Part 1/2

This is part – I of our article on Factors. In this article, we will understand how to calculate the number of factors of a number and the generalized formula for the same.

Factors – What is it?

To start our discussion, we need to know what are factors. Factors (also known as divisors) are those numbers that completely divide the original number.

Suppose we say that: ‘a’ is a factor of ‘b’, then what does this mean?

• It means that the number ‘a’ completely divides the number ‘b’ leaving no remainder.
• It can also be expressed as

b/a = k (Where k is a positive integer)

Let us take an example to understand this.

If we ask you, what are the factors of 10? What will be your response?

• Since factors of a number are those numbers that divide the given number without leaving any remainders, factors of 10 will be 1, 2, 5, and 10.
• If you observe all these numbers i.e., 1, 2, 5, and 10 will leave no remainder when divided by 10.

10 = 1 x 10 + 0

10 = 2 x 5 + 0

10 = 5 x 2 + 0

10 = 1 x 10 + 0

Example 2:

Similarly, if we ask you, what are the factors of 24? What will be your response?

• We can use the same logic and say that the factors of 24 will be 1, 2, 3, 4, 6, 8, 12, and 24.

The number of factors – Easy to find?

Now that we have understood what are ‘factors’, let’s move to another important question, “How do we calculate the number of factors for a number?”

• For instance, in our earlier examples, we saw that
  • 10 has 4 factors – 1. 2, 5 and 10
  • 24 has 8 factors – 1, 2, 3, 4, 6, 8, 12 and 24

But what if we ask you, “How many factors does 120 have?”

• One way is to write down all the factors and count them manually.
  • But as you must have already realized that it will be a tedious process to find out the number of factors of a number such as ‘120’ or ‘432’.
• So, is there any alternate way to find this out?
  • Yes, there is one. To understand the logic behind that rule let us understand how we get all the factors of the number 24.
    
    

How the factors are formed?

To begin with, we need to prime factorize the number 24.

24 = 2³ x 3¹

Now, let’s understand the importance of prime factorization.

Let us try to answer a simple question – What are the different powers of 2, that will divide this number completely?

• One may tend to think those are 2¹, 2², and 2³.
  • However, if you look carefully, you will understand 2⁰ is another power that will completely divide this number without leaving any remainder.
• This means the different powers of 2 will be 2⁰, 2¹, 2², and 2³.

In a similar line, our next target is to identify the different powers of 3, which will divide this number completely.

• You should be able to see that these are 3⁰ and 3¹

Now you may wonder how these prime factors are important to understand the formula for the number of factors.

• All the factors of 24 are formed by different combinations of powers of 2 multiplied by powers of 3.
  • Let’s look at the grid to understand it better.

For example

• 1 is formed by 2⁰ x 3⁰
• 12 is formed by 2² x 3¹
• Similarly, other factors are also formed.

So, all we need to do here is to multiply the number of rows with the number of columns.

• And how do we know the number of rows and columns?
  • The number of rows here is “the ‘power of 3’ + 1”
  • The number of columns here is “the ‘power of 2’ + 1”

So, when we know that, 24 = 2³ ×3¹, the number of factors will be (3 + 1) × (1 + 1) = 4 × 2 = 8.

General formula

So, to generalize the formula for the total number of factors,

If a number N can be prime factorized as

N = (P₁)^a x (P₂)^b x (P₃)^c…..

(Where, P₁, P₂ , P_3, and so on are distinct prime numbers) then the

total number of factors =  (a + 1) x (b + 1) x (c+1) …..

Let’s now answer the question we asked earlier. “How many factors does 120 have?”

Solution:

120 can be prime factorized as 120 = 2³ x 3¹ x 5¹

Thus, the total number of factors of 120 = (3 + 1) x (1 + 1) x (1 + 1) = 4 x 2 x 2 = 16

This is all for this part. In the next part, we will discuss how to find out the number of odd factors and even factors of a number.