Exploring Divisibility and Repunits in Number Sequences

In the sequence $1, 11, 111, 1111, \ldots$:

1. Prove that there exist two terms in the sequence whose difference is divisible by $196,673$.
2. Deduce that there exists a repunit that is divisible by $196,673$.
3. For any natural number $a$ that is not divisible by $2$ or $5$, prove that there exists a natural number $b$ such that the product $ab$ is a repunit (a number written using only the digit $1$).