[ Decimal numeral system ]

What is the smallest integer of the form 111...11 that is divisible by 333...33 (100 threes)?
Factorize the number consisting of threes.

## Solution

3... $3=3 \cdot 1 . .1$ (one hundred ones). Let the number consisting of $k$ ones be divisible by both factors. From the process of long division, it is clear that $k$ is divisible by 100. Moreover, from the divisibility rule for 3, it follows that $k$ is a multiple of 3.

## Answer

$1 \ldots 1$ (300 ones).