We can easily create a set of powers, between $1$ and $r$, inclusive:
$$\{x^1, x^2, \dots, x^r\}$$
We seek to minimize these values modulo a composite $c$ such that
$$c = q_1 \cdot q_2 \cdots q_n$$
Note that we will assume that when we say $z$ modulo $p$ for some value $p$, we mean the smallest positive integer that satisfies this statement, like in computer science.
Now, we create $r^n$ different sets by selecting one of the $r$ roots of unity modulo each prime $p$, and setting $x$ equal to this root, for each prime. Since there are $n$ primes that we use, and we pick one of the $r$th roots of unity for each prime, this gives us a total of $r^n$ possible sets.
My first observation is that there must be a power of $x$, say $x^p$, in each possible set that is greater than $(1/r) c$. In other words, we must have a value in each set such that
$$x^p > \left( \frac{1}{r} \right)(q_1 \cdot q_2 \cdots q_n)$$
This seems to happen no matter how many primes we use and how large $r$ is.
My second observation is that, if any $x^p > c/2$, we can replace this value by subtracting, i.e. if $x^p > c/2$, we replace $x^p$ with $(c-x^p)$. In other words, if $c = 100$ and $x^p = 62 \bmod 100$, then rewrite this number in the set as $(100 - 62) = 38$. So we replace large positive integers by smaller integers. If we do this, it seems that we can make the largest $x$ power arbitrarily small if we use enough primes. I seek to prove this.
EDIT
We assume that $r$ is fixed.