sequence = 0 (mod n) or else some value (mod n) is duplicated. Assume the latter, with x_a and x_b, x_b>x_a, possesing the duplicated remainders. We then have that x_b - x-a = 0 (mod n). Let m be the highest power of 10 dividing x_b - x_a. Now since (10,n) = 1, we can divide by 10^m and get that (x_b - x_a)/10^m = 0 (n). But (x_b - x_a)/10^m is a number containing only the digit 1.