Factors

Suppose we have a number n = a^p * b^q * c^r, where a, b, c are prime factors of n each raised to powers p, q and r, then,
      
Number of factors of n = (p+1)*(q+1)*(r+1)

n can be expressed as a product of 2 factors in ½*(p+1)*(q+1)*(r+1) ways.

If n is a perfect square, then it can be expressed as a product of 2 different factors in   ½*(p+1)*(q+1)*(r+1) + 1 ways.

       Sum of all factors = [{a^ (p+1) – 1}/ {a-1}] * [{b^ (q+1) – 1}/ {b-1}] * [{c^(r+1)– 1}/ {-1}]

       Number of co-primes of n = C (n) = n * (1 - 1/a) * (1 - 1/b) * (1 - 1/c)

       Sum of co-primes = [n * C (n)] /2

       Number of ways of writing n as a product of 2 co-prime numbers = 2^ [{(p+1)(q+1)(r+1)}-1]

       Product of factors = n^ [{(p+1)(q+1)(r+1)}/2]

       Number of odd & even factors of n:

Let n = (a^p * b^q * c^r) * 2^z.

       Then total number of odd factors = (a+1) (b+1) (c+1).

       Total number of even factors = Total number of factors – Total number of odd factors.

       The number of ways of expressing a composite numbers as a product of two factors = ½ * (total number of factors).