Hi Students, Do you want to learn * How To Find Total Number of Factors of Any Number*? Then, read this post of

**Amans Maths Blogs (AMB)**, in this article, you will learn about the

*, which is used to find the*

**FACTOR FORMULA****number of factor of any number**. But before, you need to about the

**fundamental theorem of arithmetic**. So lets start…

# Fundamental Theorem of Arithmetic

According to fundamental theorem of arithmetic, every positive integer can be expressed as the product of primes in a unique way. OR, you can also say that a number can be resolved into prime factors in only one way.

It mean, for a positive integer N > 1 can be uniquely written as,

^{a}× β

^{b}× γ

^{c}× δ

^{d }× …

where α < β < γ < δ < … are prime numbers and a, b, c, d, … are integer greater than or equal to 0. This decomposition of N is also known as prime factorization of N.

Let N denotes a number whose prime factorization

N = p_{1}p_{2}p_{3}…, where p_{1, }p_{2, }p_{3, }… are prime numbers.

Suppose N is also decomposed as N = q_{1}q_{2}q_{3}…, where q_{1, }q_{2, }q_{3, }… are other prime numbers.

Then, p_{1}p_{2}p_{3}… = q_{1}q_{2}q_{3}…,

Now, p1 divides the product of p_{1}p_{2}p_{3}… and since each of the factors of this product is a prime, therefore q_{1 }divides one of them p_{1}p_{2}p_{3}…,

Let q_{1 }divides one of them p_{1}. But p_{1 }and q_{1 }are prime, therefore p_{1 }and q_{1 }are equal.

Thus, p_{1}p_{2}p_{3}… = q_{1}q_{2}q_{3}…, ⇒ p_{2}p_{3}… = q_{2}q_{3}…,

Similarly q_{2 }divides one of them p_{2}p_{3}…,

Let q_{2 }divides one of them p_{2}. But p_{2 }and q_{2 }are prime, therefore p_{2 }and q_{2 }are equal.

Therefore, the prime factorization p_{1}p_{2}p_{3}… are q_{1}q_{2}q_{3}… same.

Hence, N can be resolved into prime factors in one way.

# Factor Formula

Let N is the number whose number of factors is to be calculated and its 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.

Now, the factors of α^{a} are 1, α^{1}, α^{2}, α^{3}, … α^{a}. Total number of factors of α^{a} is (a + 1).

The factors of β^{b} are 1, β^{1}, β^{2}, β^{3}, … β^{b}. Total number of factors of β^{b} is (b + 1).

Similarly,

Total number of factors of γ^{c} is (c + 1).

Total number of factors of δ^{d} is (d + 1).

Thus,

^{a}× β

^{b}× γ

^{c}× δ

^{d }× … is

(a + 1)(b + 1)(c + 1)(d + 1)…

This total number of factors of N includes 1 and the number N itself.

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

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

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

Thus, Total number of factors of 120 is (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16.

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

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

Thus, Total number of factors of 84 is (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12.

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

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

Thus, Total number of factors of 3600 is (4 + 1)(2 + 1)(2 + 1) = 5 × 3 × 3 = 45.

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

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

Thus, Total number of factors of 504 is (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24.

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

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

Thus, Total number of factors of 180 is (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18.

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

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

Thus, Total number of factors of 6480 is (4 + 1)(4 + 1)(1 + 1) = 5 × 5 × 2 = 50.

**Ques 7** : What is the number of 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 factors of 420 is (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24.

## 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 |