  We estimate exponential sums with the Fermat-like quotients $$ f_g(n) =
\frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$
are positive integers, $n$ is composite, and P(n) is the largest prime factor
of $n$. Clearly, both $f_g(n)$ and $h_g(n)$ are integers if $n$ is a Fermat
pseudoprime to base $g$, and if $n$ is a Carmichael number this is true for all
$g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts
$\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on average over $g$
for $f_g(n)$, and individually for $h_g(n)$. We also obtain similar results
with the functions ${\widetilde f}_g(n) = gf_g(n)$ and ${\widetilde h}_g(n) =
gh_g(n)$.
