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Yes, there are such numbers. Start with N, analyzing its digits from right to left. Each time you find a zero (say, at the Mth position) add N times 10^M; this will wipe out the zero (since N doesn't end in zero) without affecting the digits to the right of the Mth place. Keep on doing this until the rightmost 2004 digits are non-zero. At that time, you'll have a multiple of N that can be expressed as X.10^2004+Y, and Y has no zeroes in it. As the sum is a multiple of N, and X*10^2004 is also a multiple of N, it follows that Y must be a multiple of N, so Y is a number as we wanted. |
