Two facts to note are:

1) Multiplies of 225 end in 00 25 50 75

2) A number is divisible by 9 if the digits sum to 9.

Only those multiples ending in 00 could have only 1's and 0's.

So, the smallest digit we can multiply 225 by to get 00 at the end is 4. That is,

225*4=900. Now, we want to find the smallest multiple of 900 that contains only 1's and 0's. Let us first focus on 9 and then tack on the 00 after. The smallest multple of 9 with only 1's and 0's is 111111111. This is a consquence of the fact that the digits must sum to 9. Now, we tack on the 00 at the end and obtain 11111111100.

11111111100