Suppose we take a number 10, we are interested in numbers less than 10 and co–prime to 10. These numbers are 1,3,7,9.

Now suppose number \(n\) is \({a^m}{b^n}{c^p}\)

These number of co–primes is generally denoted by Euler’s function \(\phi (n)\).

Then number of positive integers less than \(n\) and co–prime to \(n\) is \(n\left( {1 - \frac{1}{a}} \right)\left( {1 - \frac{1}{b}} \right)\left( {1 - \frac{1}{c}} \right)...\)

Sum of all these numbers = \(\frac{{n\times\phi (n)}}{2}\)

Number of ways in which a number can be represented as sum of two numbers which are co–primes to each other is\(\frac{{\phi (n)}}{2}\)

Example 20: Find number of positive integers which are less than 360 and co–prime to 360.

Solution: The number 360 can be written as 233251

\(\phi (360)\)=\(360.\left( {1 - \frac{1}{2}} \right)\left( {1 - \frac{1}{3}} \right)\left( {1 - \frac{1}{5}} \right)\)=96