Let A denote the smallest positive integer that is divisible by 36, and whose base-10 representation consists of only 4s and 9s, with at least one of each. Find A

Answer 1

If something is divisible by 36, then it has to be divisible by both 9 and 4.

Consider first that it's divisible by 4. Then the last 2 digits have to be 44, so that the number is divisible by 4. Then, the number has to be divisible by 9. In order to be divisible by 9, the number has to have a sum of digits that's a multiple of 9. Clearly, adding nines isn't going to change that property, because 9 is already a multiple of 9. So that means the smallest number divisible by 36 is going to be 444444444 (nine 4s). However, there also has to be at least one of each - which means that we will end up with 4444444944 as our answer.