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Hint to puzzle 81: Digit transfer

Suppose the n-digit integer s = a1a2a3...an is multiplied by k when the digit an is transferred to the beginning of the number.  (That is, t = ana1a2...an−1 = ks, where t is the resulting number.)

Consider the infinite repeating decimals x = 0.a1a2a3...ana1a2a3...an... and y = 0.ana1a2...an−1ana1a2...an−1... , formed by repeating s and t, respectively.