Can you find some patterns relating the denominator of a fraction to the number of repeating digits in its decimal?

Don and these students were involved in looking at the decimals for certain fractions: MaggieP, KatieR, & VittoriaD. The decimals were obtained using Mathematica®. Maggie P, did the most thinking about the data below.

# 95564005069708491762

denominator = 789, number of repeating digits =none.  Not enough digits.

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N[85/789,1000]

0.1077313054499366286438529784537389100126742712294043092522179974651457541191381495

564005069708491761723700887198986058301647655259822560202788339670468948035487959442

332065906210392902408111533586818757921419518377693282636248415716096324461343472750

316856780735107731305449936628643852978453738910012674271229404309252217997465145754

119138149556400506970849176172370088719898605830164765525982256020278833967046894803

548795944233206590621039290240811153358681875792141951837769328263624841571609632446

134347275031685678073510773130544993662864385297845373891001267427122940430925221799

746514575411913814955640050697084917617237008871989860583016476552598225602027883396

704689480354879594423320659062103929024081115335868187579214195183776932826362484157

160963244613434727503168567807351077313054499366286438529784537389100126742712294043

092522179974651457541191381495564005069708491761723700887198986058301647655259822560

202788339670468948035487959442332065906210392902408111533586818757921419518378

denominator = 789 = 3x263, number of repeating digits = 262

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N[55/34,100]

1.61764705882352941176470588235294117647058823529411764705882352941176470588235294

1176470588235294118

denominator = 34=2x17, number of repeating digits=16

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N[1/34,100]

0.02941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941

denominator = 34=2x17, number of repeating digits=16

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N[20/34,100]

0.588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588

denominator = 34=2x17, number of repeating digits=16

N[31/34,100]

0.911764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529412

denominator = 34=2x17, number of repeating digits=16

N[67/81,100]

0.827160493827160493827160493827160493827160493827160493827160493827160493827160493827160493827160494

denominator = 81=9x9, number of repeating digits=9

N[ 9/81,100]

0.1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

denominator = 81, fraction reduces to 1/9; number of repeating digits=1

N[ 10/81,100]

0.1234567901234567901234567901234567901234567901234567901234567901234567901234567901234567901234567901

denominator = 81=9x9, number of repeating digits=9 (curious 9 digits!)

N[ 10/91,100]

0.1098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901099

denominator = 91=13x7, number of repeating digits=6

N[ 11/91,100]

0.1208791208791208791208791208791208791208791208791208791208791208791208791208791208791208791208791209

denominator = 91=13x7, number of repeating digits=6

N[ 12/91,100]

0.1318681318681318681318681318681318681318681318681318681318681318681318681318681318681318681318681319

denominator = 91=13x7, number of repeating digits=6

N[ 13/91,100]

0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571429

denominator = 91=13x7, fraction reduces to 1/7; number of repeating digits=6

N[2/91,100]

0.02197802197802197802197802197802197802197802197802197802197802197802197802197802197802197802197802198

denominator = 91=13x7, number of repeating digits=6

N[ 11/70,100]

0.1571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571

denominator = 70=2x5x7, number of repeating digits=6

N[10/92,100]

0.1086956521739130434782608695652173913043478260869565217391304347826086956521739130434782608695652174

denominator = 92=2x2x23, fraction reduces to 5/(2x23); number of repeating digits=22

N[11/92,100]

0.1195652173913043478260869565217391304347826086956521739130434782608695652173913043478260869565217391

denominator = 92=2x2x23, number of repeating digits=22

N[109/92,100]

1.184782608695652173913043478260869565217391304347826086956521739130434782608695652173913043478260870

denominator = 92=2x2x23, number of repeating digits=22

N[1/33,100]

0.03030303030303030303030303030303030303030303030303030303030303030303030303030303030303030303030303030

denominator = 33=11x 3, (1/33=3/99=.0303…); number of repeating digits=2

N[1/47,100]

0.02127659574468085106382978723404255319148936170212765957446808510638297872340425531914893617021276596

denominator = 47 (prime); number of repeating digits=46

N[1/7,100]

0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571429

denominator = 7 (prime); number of repeating digits=6

N[1/14,100]

0.0714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714

denominator = 14=2x 7; number of repeating digits=6

N[1/12,100]

0.0833333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

denominator = 12=3x2x2; number of repeating digits=1

N[2/9,100]

0.2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222

denominator = 9=3x3; number of repeating digits=1

N[15/99,100]

0.1515151515151515151515151515151515151515151515151515151515151515151515151515151515151515151515151515

denominator = 99=11x3x3; number of repeating digits=2

N[137/999,100]

0.1371371371371371371371371371371371371371371371371371371371371371371371371371371371371371371371371371

denominator = 999=37x3x3x3; number of repeating digits=3

N[1/51,100]

0.01960784313725490196078431372549019607843137254901960784313725490196078431372549019607843137254901961

denominator = 51=3x17; number of repeating digits=16

N[87/678888888,100]

1.28150572999215152863129496383802941255388466455500447077578326779153292018531315245212851384894077x10 –7

What’s your guess on this one?

Maggie’s remarks:

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