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 FACTOR FORMULA, which is used to find the 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,
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 = p1p2p3…, where p1, p2, p3, … are prime numbers.
Suppose N is also decomposed as N = q1q2q3…, where q1, q2, q3, … are other prime numbers.
Then, p1p2p3… = q1q2q3…,
Now, p1 divides the product of p1p2p3… and since each of the factors of this product is a prime, therefore q1 divides one of them p1p2p3…,
Let q1 divides one of them p1. But p1 and q1 are prime, therefore p1 and q1 are equal.
Thus, p1p2p3… = q1q2q3…, ⇒ p2p3… = q2q3…,
Similarly q2 divides one of them p2p3…,
Let q2 divides one of them p2. But p2 and q2 are prime, therefore p2 and q2 are equal.
Therefore, the prime factorization p1p2p3… are q1q2q3… 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 = αa × βb × γc × δd × …, where α, β, γ, δ, … are prime numbers and a, b, c, d, … positive integers.
Now, the factors of αa are 1, α1, α2, α3, … αa. Total number of factors of αa is (a + 1).
The factors of βb are 1, β1, β2, β3, … βb. Total number of factors of βb is (b + 1).
Similarly,
Total number of factors of γc is (c + 1).
Total number of factors of δd is (d + 1).
Thus,
(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 = 23 × 31 × 51.
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 = 22 × 31 × 71.
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 = 24 × 32 × 52.
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 = 23 × 32 × 71.
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 = 22 × 32 × 51.
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 = 24 x 34 x 51.
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 = 22 x 31 x 51 x 71.
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 |
