Find the smallest positive integer $n$ such that every digit of $15n$ is either $8$ or $0$.
This is one of the questions we presented in one session to contest preparation PUTNAM. It turns out that I can't get from the problem. Could someone just give me a hint? (Please, don't give me the answer. Simply, an argument that can help me advance in the problem or theorem might suffice.)
Best Answer
You could try brute force with multiples of 15 but that might be too slow. You know that you are dealing with numbers consisting of only $0$s and $8$s. You also know that your number must be divisible by $15$ so it should end in $0$ and have $8$s in multiples of $3$.
You didn't want the answer but I can tell you that the smallest number with this property works.