Find the sum of all positive irreducible fractions less than 1 whose denominator is 2016. I know that we first have to use AP to find the sum of first 2015 natural nos. Then we need to subtract all those factors of 2016. But there are too many. I need help in this part.

The prime factors of $2016$ are $2$, $3$ and $7$.
Hence use inclusion/exclusion principle:


*

*Include $\sum\limits_{n=1}^{2016}\frac{n}{2016}$

*Exclude $\sum\limits_{n=1}^{\frac{2016}{2}}\frac{2\cdot{n}}{2016}$

*Exclude $\sum\limits_{n=1}^{\frac{2016}{3}}\frac{3\cdot{n}}{2016}$

*Exclude $\sum\limits_{n=1}^{\frac{2016}{7}}\frac{7\cdot{n}}{2016}$

*Include $\sum\limits_{n=1}^{\frac{2016}{2\cdot3}}\frac{2\cdot3\cdot{n}}{2016}$

*Include $\sum\limits_{n=1}^{\frac{2016}{2\cdot7}}\frac{2\cdot7\cdot{n}}{2016}$

*Include $\sum\limits_{n=1}^{\frac{2016}{3\cdot7}}\frac{3\cdot7\cdot{n}}{2016}$

*Exclude $\sum\limits_{n=1}^{\frac{2016}{2\cdot3\cdot7}}\frac{2\cdot3\cdot7\cdot{n}}{2016}$

