  We show, in an effective way, that there exists a sequence of congruence
classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the
congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log
m_k\to\infty $ as $k\to\infty$. Here, $\phi(n)$ is the Euler function. This
answers a question raised in \cite{FS}. We also show that every congruence
class containing an even integer contains infinitely many values of the
Carmichael function $\lambda(n)$ and the least such $n$ satisfies $n\ll
m^{13}$.
