1 and 5 constitue a reduced residue system (mod 6).
The book says a set of integers $a_1,...,a_h$ is a reduced residue system if it's incongruent (mod m) and relatively prime to m, such that if a is any integer prime to m, there is an index i, $1 \leq i \leq h$, for which $a \equiv a_i(mod$ $ m)$.
1 and 5 are incongruent to (mod 6). Each one is relatively prime to 6. But what bothers me is that for some integers a, the options won't satisfy $a \equiv a_i(mod$ $ m)$.
Community wiki answer so the question can be marked as answered:
As discussed in the comments, the premise that $a$ is prime to $m$ is essential, and indeed any integer $a$ prime to $m$ either $a\equiv1\bmod6$ or $a\equiv5\bmod6$.