3-2. Find the smallest number, written with only ones, that is divisible by the number composed of a hundred threes ( \(333 \ldots 33\) ).
Solve 2. Answer: \(\underbrace{11 \ldots 1}_{300}\). The number \(a_{n}=\underbrace{11 \ldots 1}_{n}\) is divisible by \(\underbrace{33 \ldots 3}_{100}\) if and only if \(n\) is divisible by 3 and \(a_{n}\) is divisible by \(a_{100}\). We will show that \(a_{n}\) is divisible by \(a_{m}\) if and only if \(n\) is divisible by \(m\). Clearly, \(9 a_{n}=10^{n}-1\). Therefore, \(a_{n}\) is divisible by \(a_{m}\) if and only if \(10^{n}-1\) is divisible by \(10^{m}-1\). Let \(n=d m+r\), where \(0 \leqslant r<m\). Then

\[
10^{n}-1=\left(10^{n-m}+10^{n-2 m}+\cdots+10^{n-d m}\right)\left(10^{m}-1\right)+10^{r}-1 .
\]

Thus, \(10^{n}-1\) is divisible by \(10^{m}-1\) if and only if \(n\) is divisible by \(m\).