Ones and Zeros - Answer

Consider the numbers 1, 11, 111, 1111, . . . up to the number with n 1's. Now consider the remainders when each of these numbers is divided by n. If any of the remainders is zero, we're done — that number is divisible by n. If none of the remainders is zero, then there are at most n-1 different remainders. But there are n numbers. So two of the remainders must be the same. Take the two numbers that had the same remainders when divided by n, subtract the smaller from the larger, and the result is a number, consisting only of zeros and ones, that is divisible by n.