Least totient in a residue class

For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with phi(n) equivalent to a (mod m). Here, as usual phi(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler function in reduced residue classes modulo m. Some discussion and results are also given for classes with gcd(a, m) > 1, in which case such n do not always exist, and also on the related problem for 'cototients'.