What's the easiest implementation for transforming a graph to its dual? I couldn't find any easy to understand algos so i need some help.

As for many combinatorial etc. tasks we choose to perform %(mod) operation with a prime no.(M) but is the information lost?i.e Can we map back to the orignal no of combinations whose max value we actually restricted to M.

Extra question:

In the given image while calculating %p for each why is np%p=1 {n=1,2,3,4.. } instead of 0.

https://www.codechef.com/problems/EDIT Can someone give me an intuition on how to approach this problem ?