Numbers the Multiple of whose Reciprocal are Cyclic Permutations

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Theorem

Let $m \in \Z_{>0}$.

Consider the reciprocal of $m$.


Let $n \in \Z$ such that $1 \le n < m$.

Then:

The digits in the decimal expansion of the rational number $\dfrac n m$ form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 m$

if and only if:

$(1): \quad m$ is the integer power of a prime number $p$
$(2): \quad$ The period of recurrence of the decimal expansion of $1 / p$ is $p - 1$
$(3): \quad p$ is not a divisor of $n$.


Proof


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Examples

Multiples of Reciprocal of $49$

The digits in the decimal expansion of the rational number $\dfrac n {49}$, for $1 \le n < 49$, form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 {49}$:


\(\ds 1 / 49\) \(=\) \(\ds 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1\) Reciprocal of 49
\(\ds 2 / 49\) \(=\) \(\ds 0 \cdotp \dot 04081 \, 63265 \, 30612 \, 24489 \, 79591 \, 83673 \, 46938 \, 77551 \, 0 \dot 2\)
\(\ds 3 / 49\) \(=\) \(\ds 0 \cdotp \dot 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75510 \, 20408 \, 16326 \, 5 \dot 3\)
\(\ds 4 / 49\) \(=\) \(\ds 0 \cdotp \dot 08163 \, 26530 \, 61224 \, 48979 \, 59183 \, 67346 \, 93877 \, 55102 \, 0 \dot 4\)
\(\ds 5 / 49\) \(=\) \(\ds 0 \cdotp \dot 10204 \, 08163 \, 26530 \, 61224 \, 48979 \, 59183 \, 67346 \, 93877 \, 5 \dot 5\)
\(\ds 6 / 49\) \(=\) \(\ds 0 \cdotp \dot 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 51020 \, 40816 \, 32653 \, 0 \dot 6\)
\(\ds 7 / 49\) \(=\) \(\ds 0 \cdotp \dot 14285 \, \dot 7 = 1 / 7\)
\(\ds 8 / 49\) \(=\) \(\ds 0 \cdotp \dot 16326 \, 53061 \, 22448 \, 97959 \, 18367 \, 34693 \, 87755 \, 10204 \, 0 \dot 8\)
\(\ds 9 / 49\) \(=\) \(\ds 0 \cdotp \dot 18367 \, 34693 \, 87755 \, 10204 \, 08163 \, 26530 \, 61224 \, 48979 \, 5 \dot 9\)
\(\ds 10 / 49\) \(=\) \(\ds 0 \cdotp \dot 20408 \, 16326 \, 53061 \, 22448 \, 97591 \, 98367 \, 34693 \, 87755 \, 1 \dot 0\)
\(\ds 11 / 49\) \(=\) \(\ds 0 \cdotp \dot 22448 \, 97591 \, 98367 \, 34693 \, 87755 \, 10204 \, 08163 \, 26530 \, 6 \dot 1\)
\(\ds 12 / 49\) \(=\) \(\ds 0 \cdotp \dot 24489 \, 79591 \, 83673 \, 46938 \, 77551 \, 02040 \, 81632 \, 65306 \, 1 \dot 2\)
\(\ds 13 / 49\) \(=\) \(\ds 0 \cdotp \dot 26530 \, 61224 \, 48979 \, 59183 \, 67346 \, 93877 \, 55102 \, 04081 \, 6 \dot 3\)
\(\ds 14 / 49\) \(=\) \(\ds 0 \cdotp \dot 28571 \, \dot 4 = 2 / 7\)
\(\ds 15 / 49\) \(=\) \(\ds 0 \cdotp \dot 30612 \, 24489 \, 79591 \, 83673 \, 46938 \, 77551 \, 02040 \, 81632 \, 6 \dot 5\)
\(\ds 16 / 49\) \(=\) \(\ds 0 \cdotp \dot 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75510 \, 20408 \, 1 \dot 6\)
\(\ds 17 / 49\) \(=\) \(\ds 0 \cdotp \dot 34693 \, 87755 \, 10204 \, 08163 \, 26530 \, 61224 \, 48979 \, 59183 \, 6 \dot 7\)
\(\ds 18 / 49\) \(=\) \(\ds 0 \cdotp \dot 36734 \, 69387 \, 75510 \, 20408 \, 16326 \, 53061 \, 22448 \, 97959 \, 1 \dot 8\)
\(\ds 19 / 49\) \(=\) \(\ds 0 \cdotp \dot 38775 \, 51020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 6 \dot 9\)
\(\ds 20 / 49\) \(=\) \(\ds 0 \cdotp \dot 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75510 \, 2 \dot 0\)
\(\ds 21 / 49\) \(=\) \(\ds 0 \cdotp \dot 42857 \, \dot 1 = 3 / 7\)
\(\ds 22 / 49\) \(=\) \(\ds 0 \cdotp \dot 44897 \, 95918 \, 36734 \, 69387 \, 75510 \, 20408 \, 16326 \, 53061 \, 2 \dot 2\)
\(\ds 23 / 49\) \(=\) \(\ds 0 \cdotp \dot 46938 \, 77551 \, 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 7 \dot 3\)
\(\ds 24 / 49\) \(=\) \(\ds 0 \cdotp \dot 48979 \, 59183 \, 67346 \, 93877 \, 55102 \, 04081 \, 63265 \, 30612 \, 2 \dot 4\)
\(\ds 25 / 49\) \(=\) \(\ds 0 \cdotp \dot 51020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 7 \dot 5\)
\(\ds 26 / 49\) \(=\) \(\ds 0 \cdotp \dot 53061 \, 22448 \, 97959 \, 18367 \, 34693 \, 87755 \, 10204 \, 08163 \, 2 \dot 6\)
\(\ds 27 / 49\) \(=\) \(\ds 0 \cdotp \dot 55102 \, 04081 \, 63265 \, 30612 \, 24489 \, 79591 \, 83673 \, 46938 \, 7 \dot 7\)
\(\ds 28 / 49\) \(=\) \(\ds 0 \cdotp \dot 57142 \, \dot 8 = 4 / 7\)
\(\ds 29 / 49\) \(=\) \(\ds 0 \cdotp \dot 59183 \, 67346 \, 93877 \, 55102 \, 04081 \, 63265 \, 30612 \, 24489 \, 7 \dot 9\)
\(\ds 30 / 49\) \(=\) \(\ds 0 \cdotp \dot 61224 \, 48979 \, 59183 \, 67346 \, 93877 \, 55102 \, 04081 \, 63265 \, 3 \dot 0\)
\(\ds 31 / 49\) \(=\) \(\ds 0 \cdotp \dot 63265 \, 30612 \, 24489 \, 79591 \, 83673 \, 46938 \, 77551 \, 02040 \, 8 \dot 1\)
\(\ds 32 / 49\) \(=\) \(\ds 0 \cdotp \dot 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 51020 \, 40816 \, 3 \dot 2\)
\(\ds 33 / 49\) \(=\) \(\ds 0 \cdotp \dot 67346 \, 93877 \, 55102 \, 04081 \, 63265 \, 30612 \, 24489 \, 79591 \, 8 \dot 3\)
\(\ds 34 / 49\) \(=\) \(\ds 0 \cdotp \dot 69387 \, 75510 \, 20408 \, 16326 \, 53061 \, 22448 \, 97959 \, 18367 \, 3 \dot 4\)
\(\ds 35 / 49\) \(=\) \(\ds 0 \cdotp \dot 71428 \, \dot 5 = 5 / 7\)
\(\ds 36 / 49\) \(=\) \(\ds 0 \cdotp \dot 73469 \, 38775 \, 51020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 3 \dot 6\)
\(\ds 37 / 49\) \(=\) \(\ds 0 \cdotp \dot 75510 \, 20408 \, 16326 \, 53061 \, 22448 \, 97959 \, 18367 \, 34693 \, 8 \dot 7\)
\(\ds 38 / 49\) \(=\) \(\ds 0 \cdotp \dot 77551 \, 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 3 \dot 8\)
\(\ds 39 / 49\) \(=\) \(\ds 0 \cdotp \dot 79591 \, 83673 \, 46938 \, 77551 \, 02040 \, 81632 \, 65306 \, 12244 \, 8 \dot 9\)
\(\ds 40 / 49\) \(=\) \(\ds 0 \cdotp \dot 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 51020 \, 4 \dot 0\)
\(\ds 41 / 49\) \(=\) \(\ds 0 \cdotp \dot 83673 \, 46938 \, 77551 \, 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 9 \dot 1\)
\(\ds 42 / 49\) \(=\) \(\ds 0 \cdotp \dot 85714 \, \dot 2 = 6 / 7\)
\(\ds 43 / 49\) \(=\) \(\ds 0 \cdotp \dot 87755 \, 10204 \, 08163 \, 26530 \, 61224 \, 48979 \, 59183 \, 67346 \, 9 \dot 3\)
\(\ds 44 / 49\) \(=\) \(\ds 0 \cdotp \dot 89795 \, 91836 \, 73469 \, 38775 \, 51020 \, 40816 \, 32653 \, 06122 \, 4 \dot 4\)
\(\ds 45 / 49\) \(=\) \(\ds 0 \cdotp \dot 91836 \, 73469 \, 38775 \, 51020 \, 40816 \, 32653 \, 06122 \, 44897 \, 9 \dot 5\)
\(\ds 46 / 49\) \(=\) \(\ds 0 \cdotp \dot 93877 \, 55102 \, 04081 \, 63265 \, 30612 \, 24489 \, 79591 \, 83673 \, 4 \dot 6\)
\(\ds 47 / 49\) \(=\) \(\ds 0 \cdotp \dot 95918 \, 36734 \, 69387 \, 75510 \, 20408 \, 16326 \, 53061 \, 22448 \, 9 \dot 7\)
\(\ds 48 / 49\) \(=\) \(\ds 0 \cdotp \dot 97959 \, 18367 \, 34693 \, 87755 \, 10204 \, 08163 \, 26530 \, 61224 \, 4 \dot 8\)


Sources

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