Number System: Factors and Coprimes

A number can be written in its prime factorization format.  For example 100 = 2² x 5² 

Formula 1: The number of factors of a number N = a^p x b^q x c^r ... = (p+1).(q+1).(r+1)...

Example: Find the number of factors of 100.  
Ans: We know that 100 = 2² x 5²
So number of factors of 100 = (2 +1 ).(2 +1) =  9.
Infact the factors are 1, 2, 4, 5, 10, 20, 25, 50, 100

Formula 2: The sum of factors of a number N=ap.bq.cr... can be written as ap+1−1a−1×bq+1−1b−1×cr+1−1c−1...

Example: Find the sum of the factors of 72
Ans: 72 can be written as (23×32). 

Sum of all the factors of 72 = (23+1−12−1×32+1−13−1)= 15 x 13= 195

Formula 3: The number of ways of writing a number as a product of two number =12×[(p+1).(q+1).(r+1)...] (if the number is not a perfect square)

If the number is a perfect square then two conditions arise:

1.  The number of ways of writing a number as a product of two distinct numbers =12×[(p+1).(q+1).(r+1)...−1]

2.  The number of ways of writing a number as a product of two numbers and those numbers need not be distinct= 12×[(p+1).(q+1).(r+1)...+1]