Suppose the number 30 is represented as product of two positive numbers. Since 30 has 8 factors.

It is visible that total number of pairs

 = \(\frac{{Number\,\,of\,factors}}{2}\) 

If the number is a perfect square, it has odd number of divisors. For example 36 has 9 divisors.

So total number of pairs having distinct factors =\(\frac{{Number\,\,of\,factor\, - \,1}}{2}\)

So total number of pairs (including distinct and same factors) =\(\frac{{Number\,\,of\,factor\, + \,1}}{2}\)

Example 17: In how many ways 900 can be represented as product of two different natural numbers?

Solution:  The number 900 can be written as \({2^2}{3^2}{5^2}\) 

Hence number of factor are 3.3.3 =27

So the number of ways = \((27 - 1)/2 =13\).