Number of Divisors
The number of divisors, referred as d(n) of a positive integer n is the total of all the positive integers that evenly divide n without leaving a remainder. Let’s under this by taking an example.
For example, the divisors of 12 are 1,2,3,4,6 and 12. Therefore, the number of divisors of 12 is d(12) = 6
Number of Divisors Formula
Number of Divisors for any given number is calculated using the following formula:
d(n) = (x1 + 1) (x2 + 1) ………. (xk + 1)
Where x1, x2, . . . xk are the exponent raised to the prime factors of n.
Let’s consider an example for better understanding.
The prime factorization of 12 is 22 × 3. Therefore, the number of divisors of 12 are:
d(12) = (2 +1) (2+1) (1+1) = 6
Euler’s Totient Function
Euler’s totient function, denoted as ϕ(n), counts the positive integers less than or equal to n that are coprime with n. In other words, they share no common factors other than 1.
Euler’s Totient Function is calculated by using the following formula:
ϕ(n) = n × (1 – 1/p1) × (1 – 1/p2) × . . . × (1 – 1/pk)
Where n = ap1 × bp2 × cp3 × . . . × kpk.
Let’s understand this by taking examples.
Example 1: For n = 12, find the number of divisors.
Solution:
Prime factorization of 12 = 22 x 3
ϕ(12) = 12 ( 1 – ½ ) (1 – ⅓) = 12 ( ½ x ⅔) = 4
Therefore, 12 has 4 positive integers less than or equal to 12 that are relatively prime to 12: 1, 5, 7, and 11
Example 2: Find ϕ(20).
Solution:
Prime factorization of 20 = 22 x 5
ϕ(20) = 20 (1 – ½ ) (1 – ⅓) = 20 ( ½ x ⅘) = 8
Therefore, 20 has 8 positive integers less than or equal to 20 that are relatively prime to 20: 1, 3, 7, 9, 11, 13, 17 and 19.
Divisors in Maths
Divisor is the number from which we divide the dividend to determine the quotient and remainder. In arithmetic, division is one of the four fundamental operations; other operations are addition, subtraction, and multiplication.
Divisors in Number Theory are integers that divides another integer without leaving the remainder is also called a divisor.
In this article, we will discuss both definitions of a divisor, including the general, and the definition in number theory. We will also explore various properties and examples related to divisors and discuss concepts such as prime divisors, the number of divisors, the sum of divisors, and the difference between a divisor and a factor.
Table of Content
- What are Divisors?
- Properties of Divisors
- Divisors and Dividends
- Divisor in Number Theory
- Examples of Divisors
- What are Prime Divisors?
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