Ken's POTW

Dividing by 111
Tests for divisibility exist for small numbers: So my puzzle is this:
  1. Find a test for divisibility by 111.
  2. Extending this, find a test for divisbility by any number consisting solely of 1's.
Please explain why your tests work.

Source: An original extension of UofCF's Problem of the Week ending 11/9/97.

Mail to Ken