Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in $$Z_n$$. Run the program for various values of $$n$$. Use these data to make conjectures about the number of idempotents in $$Z_n$$ as a function of $$n$$. For example, how many idempotents are there when $$n$$ is a prime power? What about when $$n$$ is divisible by exactly two distinct primes? In the case where $$n$$ is of the form $$pq$$ where $$p$$ and $$q$$ are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given $$n$$ and the number of distinct prime divisors of $$n$$?

Please enter $$n$$ and click the button, the idempotents in $$Z_n$$ will show below.