Hi Students, Welcome to **Amans Maths Blogs (AMB)**. As you have read my post * How To Find Total Number of Factors of Any Number*? So, you know about how to find total numbers of any given number. In those factors some of them are EVEN factors and others are ODD factors. In this article, you will learn other

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**FACTOR FORMULA***?. In the other article, you will learn about*

**How To Find Number of EVEN Factors of Any Number****how to find number of ODD factors of any given number**.

# Factor Formula

To understand what is going to be find, lets start with an example.

Suppose you are given a number 150.

If N is the number whose prime factorization is

N = α^{a} × β^{b} × γ^{c} × δ^{d }× …, where α, β, γ, δ, … are prime numbers and a, b, c, d, … positive integers.

Then the total number of factors of N is (a + 1)(b + 1)(c + 1)(d + 1)….

Here, you are given N = 150 (an even number).

Then, its **prime factorization is 150 = ****2 × 3 × 5 ^{2}.** …………………………………………(1)

Thus, Total number of factors of 150 is (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12.

Now, the factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

In these factors, you found that 2, 6, 10, 30, 50, 150 are the EVEN factors of 150. There are only 6 even factors of 150.

Let another N = 6655 (an odd number).

Then, its **prime factorization is 6655 = ****5 × 11 ^{3}.** ……………………………………..(2)

Thus, Total number of factors of 6655 is (1 + 1)(3 + 1) = 2 × 3 = 6.

Now, the factors of 6655 are 1, 5, 11, 55, 121, 605, 1331, 6655.

In these factors, you found that there is no EVEN factors of 6655.

Since 150 and 6655 is not a large number, so you can find even factors of 150 and 6655 easily by listing all the factors.

BUT, if you are given a large number whose the number is factors is large, then you find that it is very difficult to find number of even factors of the number.

Therefore, here is given factor formula, using that you can find the number of even factors of the number, without listing all the factors.

Have you observed that 150 has 6 even factors whereas 6655 has no even factors? Why does this happen?

Observe the prime factorizations of 150 and 6655.

Since the prime factorization of 150 contains the prime factor 2, so 150 has even factor.

whereas, the prime factorization of 6655 does not contain the prime factor 2, so 6655 has no even factors.

Thus, the number of even factors depends on the prime factor 2 of prime factorization of any number.

It means only an even number has even factors whereas an odd number has no even factors.

Now, understand the factor formula below to find the number of even factors of any number.

## How to Find Total Number of EVEN Factors of a Number

Let N is the number whose the prime factorization is N = α^{a} × β^{b} × γ^{c} × δ^{d }× …, where α < β < γ < δ < … are prime numbers and a, b, c, d, … positive integers.

^{a}× β

^{b}× γ

^{c}× δ

^{d }× … is

**(a)(b + 1)(c + 1)(d + 1)…**, if α = 2 (N is an EVEN number)

^{a}× β

^{b}× γ

^{c}× δ

^{d }× … is

**0**, if α

**≠**2 (N is an ODD number)

**Ques 1** : Find the total number of even factors of 120.

**Solution** : Prime Factorization of 120 is 120 = 2^{3} × 3^{1} × 5^{1}.

Thus, Total number of even factors of 120 is (3)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12.

**Ques 2** : Find the total number of even factors of 84.

**Solution** : Prime Factorization of 84 is 84 = 2^{2} × 3^{1} × 7^{1}.

Thus, Total number of even factors of 84 is (2)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8.

**Ques 3** : What is the number of even factors of the number 3600?

**Solution** : Prime Factorization of 3600 is 3600 = 2^{4} × 3^{2} × 5^{2}.

Thus, Total number of even factors of 3600 is (4)(2 + 1)(2 + 1) = 4 × 3 × 3 = 36.

**Ques 4** : What is the number of even factors of the number 504?

**Solution** : Prime Factorization of 504 is 504 = 2^{3} × 3^{2} × 7^{1}.

Thus, Total number of even factors of 504 is (3)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18.

**Ques 5** : Find the total number of even factors of 180.

**Solution** : Prime Factorization of 180 is 180 = 2^{2} × 3^{2} × 5^{1}.

Thus, Total number of even factors of 180 is (2)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12.

**Ques 6** : What is the number of even factors of 6480?

**Solution** : Prime Factorization of 6480 is 6480 = 2^{4} x 3^{4} x 5^{1}.

Thus, Total number of even factors of 6480 is (4)(4 + 1)(1 + 1) = 4 × 5 × 2 = 40.

**Ques 7** : What is the number of even factors of 420?

**Solution** : Prime Factorization of 420 is 420 = 2^{2} x 3^{1} x 5^{1 }x 7^{1}.

Thus, Total number of even factors of 420 is (2)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16.

## Factors and Multiples Table

Formula for Total Factors | Formula for Even Factors |

Formula for Odd Factors | Formula for Sum of All Factors |

Formula for Product of All Factors | Factors & Multiples |