What is the smallest multiple of 45 that only has 1 and 0 as digits?

You can experiment a bit for this one. Or you can apply some reasoning:* Multiples of 45 end with 5 or 0. To avoid a 5, you would have to multiply by an even number; that is, you are looking for multiples of 90.

* Since multiples of 90 are also multiples of 9, the sum of the digits must needs be 9.

This means that you must provide 9 ones (or a multiple thereof), and the last digit must be 0. The smallest number that satisfies these criteria is 1111111110.