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Liking my Threes Puzzle (Tanton Mathematics)

Here is a cute puzzle for those who have a predilection for the number three. Is there a number N so that it and all its multiples 2N, 3N, 4N, ... each contain the digit 3? N=65913 gives a good run: N, 2N, 3N, .., 10N each contain the digit three, but 11N does not. Is there a "perfect" N with an infinitely long run of success?
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