Numbers Composed of Only 1's

Part 1. Consider the numbers a and b such that a = 1111...1111 (m times) and b = 1111...1111 (n times). Prove that a is divisible by b if and only if m is divisible by n, assuming with no loss of generality that a >= b and m >= n.

Part 2. Find the smallest number composed of only 1's that is divisible by 3333...3333 (100 3's). Prove, using the result of part 1, that the number is indeed the smallest such number.

My Answer to Part 2 - the number of 1's in this number is:

Solution

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