Question 36
Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: For any positive integer $d \geq 2$, when $S$ is interpreted as a number in base $d$, the resulting number is non-zero and divisible by $n$.
Remark: The sequence $S=s_{k} s_{k-1} \cdots s_{1} s_{0}$ interpreted in base $d$ is the number $\sum_{i=0}^{k} s_{i} d^{i}$.
Remark: The sequence $S=s_{k} s_{k-1} \cdots s_{1} s_{0}$ interpreted in base $d$ is the number $\sum_{i=0}^{k} s_{i} d^{i}$.