What's the smallest positive integer made of 1's that is divisible by a number made out of 100 9's

I was asked what is the smallest positive integer made of ones (11111...1) that is divisible (no reminder) by a number made up of 100 digits of 9 (9999...9).

I noticed that for 9 the smallest integer made out of ones that will divisible by him, is 111 111 111 (9 digits), and that for 99 the integer will be with 111 111 111 111 111 111 (9*2 digits).

From this I assume that the answer will be 9*100 digits of 1.

So my question is first is this true and second how can I prove it.

Thanks.

• Do you mean THE number made by 100 digits of 9? – quanta Apr 2 '11 at 16:28
• Hint: Divisibility by the number which has 100 digits of 9 is equivalent to divisibility by 9 and by the number which has 100 digits of 1 (since casting-out-nines shows these are coprime). Now apply these restrictions to a number that is a string of 1's. – hardmath Apr 2 '11 at 16:36

Hint: express the chain of $k$ $1$'s as $\frac{10^n-1}{9}$ (how do $k$ and $n$ relate?) and the $9$'s similarly, and think of factoring them.