The integers coprime to $n$ are called the {\it totatives} \rm of $n$. D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of the total number of totatives up to $n$. They provided criteria in various cases. Here we give an ``if and only if'' criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character $\chi \pmod \kappa$ (where $\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$ dividing $n$ for which $\chi(p)=1$.

D.~H.~Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P.~Erd\H os, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

We prove that $\{(n^p-n)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$ ranging over all primes, is independent of $1$ over the integers, assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.