2017 IMO Shortlist N4

Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \in\{1,2,3, \ldots, 2017\}$ such that $\frac{10^{t}-1}{c \cdot m}$ is short, and such that $\frac{10^{k}-1}{c \cdot m}$ is not short for any $1 \leq k<t$. Let $S(m)$ be the set of $m$-tastic numbers. Consider $S(m)$ for $m=1,2, \ldots$. What is the maximum number of elements in $S(m)$ ? (Turkey)

如果一个有理数的十进制展开式中位数有限，则称该有理数为短数。对于正整数 $m$，如果存在一个数字 $c \in\{1,2,3, \ldots, 2017\}$ 使得 $\frac{10^{t}-1}{c \cdot m}$ 很短，并且 $\frac{10^{k}-1}{c \cdot m}$ 对于任何 $1 \leq k<t$ 都不短，我们就说正整数 $t$ 是 $m$-tastic。令 $S(m)$ 为 $m$-tastic 数的集合。考虑 $S(m)$ 为 $m=1,2, \ldots$。 $S(m)$ 中的最大元素数是多少？ (火鸡)