You are thinking "given N, what is the probability that N is a perfect nth power?". Think about it in the reverse. "given N, what is the probability that the

You are thinking "given N, what is the probability that N is a perfect nth power?". Think about it in the reverse. "given N, what is the probability that the nth root of N is an integer". For a whole bunch of different Ns, we will get some real number between 0 and 1. To match up to your example. "For a number M, m is the remainder of M: m = M - floor(M), what is the probability that m is zero given M"