Factors or divisors

02 min read

For a number N, all the numbers, including 1 and N itself, which divide N completely are called divisors/factors of N. For example, the number 24 is divisible by 1, 2, 3, 4, 6, 8, 12 and 24. Hence all these numbers are divisors of 24. Also, these numbers are factors of 24.

 

Number of factors

Let N be a composite number such that N = (xa)(yb)(zc)... where x, y, z... are prime factors. N can be written as (x0 + x1 + x2...a+1 terms)(y0 + y1 + y2....b+1 terms)...

Perfect squares have an odd number of factors. Then, the number of divisors of N = (a+1)(b+1)(c+1)...

 

Even and odd factors

Using the same logic as above, Let N = (2a)(yb)(zc)... where y, z are prime factors. Then, the number of even divisors of n = (a)(b+1)(c+1)... and number of odd divisors of N = (b+1)(c+1).

 

Sum of Factors

Let N be a composite number such that N = (xa)(yb)(zc)... where x, y, z... are prime factors. Then, the sum of divisors of n = [(x(a+1) - 1)/(x-1)] x [(y(b+1) - 1)/(y-1)]...

 

Product of divisors

Let N be a composite number such that N = (xa)(yb)(zc)... where x, y, z... are prime factors. Then, the product of divisors of N = N[(a+1)(b+1)(c+1)/2]. In case of N being a perfect square, the product of divisors of N = [{(a+1)(b+1)(c+1)+1}/2].

 

Totient Function - Number of Numbers less than and prime to a given number

The totient φ(n) of a positive integer n greater than 1 is defined to be the number of positive integers less than n that are coprime to n. φ(1) is defined to be 1. For N = (xa)(yb)(zc)....


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