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I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime

1/prime matches the last digit of the prime number itself for several primes. This pattern holds for many primes like 7, 11, 13, 17, 19, and so on, as listed in the above examples.

Is there a mathematical explanation for this pattern, or is it a coincidence? Does this pattern have a name, or has it been studied before? How can this pattern be mathematically explained, and does it hold for all primes (except 2 and 5 which do not yield repeating decimals)?

Any insights or references to relevant mathematical literature would be greatly appreciated.

3:   1/3   = 0.3          <<<
7:   1/7   = 0.142857     <<<  
11:  1/11  = 0.09
13:  1/13  = 0.076923     <<<
17:  1/17  = 0.0588235294117647     <<<
19:  1/19  = 0.052631578947368421
23:  1/23  = 0.0434782608695652173913     <<< notice the last digit 3 matches again
29:  1/29  = 0.0344827586206896551724137931
31:  1/31  = 0.032258064516129
37:  1/37  = 0.027 <<< last digits 7 (prime) and 7 (repeating number)
41:  1/41  = 0.02439
43:  1/43  = 0.023255813953488372093
47:  1/47  = 0.0212765957446808510638297872340425531914893617
53:  1/53  = 0.0188679245283018867924528301886792452830188679
59:  1/59  = 0.0169491525423728813559322033898305084745762711864406779661
61:  1/61  = 0.016393442622950819672131147540983606557377049180327868852459
67:  1/67  = 0.014925373134328358208955223880597014925373134328358209
71:  1/71  = 0.0140845070422535211267605633802816901408450704225352112676056338028169014084507
73:  1/73  = 0.0136986301369863013698630136986301369863013698630136986
79:  1/79  = 0.01265822784810126582278481012658227848101265822784810126582278481
83:  1/83  = 0.0120481927710843373493975903614457831325301204819277108433734939759036144578313
89:  1/89  = 0.0112359550561797752808988764044943820224719101123595505617977528089887640449438
97:  1/97  = 0.010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
101: 1/101 = 0.00990099009900990099009900990099009900990099
103: 1/103 = 0.00970873786407766990291262135922330097087378640776699029126213592233009708737864077669902912621359
107: 1/107 = 0.00934579439252336448598130841121495327102803738317757009345794392523364485981308411214953271028037
109: 1/109 = 0.009174311926605504587155963302752293577981651376146789
113: 1/113 = 0.008849557522123893805309734513274336283185840707964601769911504424778761061946902654867256637168141
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    $\begingroup$ Hi, welcome to Math SE. Hint: for prime $p\notin\{2,\,5\}$ write $\frac{1}{p}=\frac{k}{10^n-1}$, so $10|kp+1$. Your observation is some $p$ satisfy $10|k-p$. Show this holds for exactly $2$ of the $4$ residue classes modulo $10$ such $p$ take. $\endgroup$
    – J.G.
    Oct 22, 2023 at 21:24
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    $\begingroup$ Some of these expansions are not terminated at the end of the first repeating block. Please fix these to make the pattern more obvious. $\endgroup$ Oct 22, 2023 at 22:48
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    $\begingroup$ You said the pattern holds for $11$ and $19$, but that doesn’t appear so $\endgroup$ Oct 22, 2023 at 22:52

1 Answer 1

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In not-very-precise language, if you multiply a prime by the repetend (repeated part of the decimal expansion), the last digit must be a 9 (so that you get an infinite string of 9s when you multiply the prime by the full decimal expansion). For example, 7 * 142857 = 999999. Now for this to happen, the last digit of the prime and the last digit of the repetend must be one of the pairs (1,9), (3,3), (7,7), (9,1). So it's the same digit if the prime ends with a 3 or 7. Is this what you are looking at?

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  • $\begingroup$ Thank you very much for the answer! Now it makes sense. $\endgroup$ Oct 24, 2023 at 12:21

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