Anomalous Cancellation on 2-Digit Numbers

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Theorem

There are exactly four anomalously cancelling vulgar fractions having two-digit numerator and denominator when expressed in base $10$ notation:


$16 / 64$

$\dfrac 1 4 = \dfrac {16} {64} = \dfrac {166} {664} = \dfrac {1666} {6664} = \cdots$


$19 / 95$

$\dfrac 1 5 = \dfrac {19} {95} = \dfrac {199} {995} = \dfrac {1999} {9995} = \cdots$


$26 / 65$

$\dfrac 2 5 = \dfrac {26} {65} = \dfrac {266} {665} = \dfrac {2666} {6665} = \cdots$


$49 / 98$

$\dfrac 4 8 = \dfrac {49} {98} = \dfrac {499} {998} = \dfrac {4999} {9998} = \cdots$


Proof


Historical Note

According to David Wells in his $1986$ work Curious and Interesting Numbers, this result, as well as some others, was demonstrated by Alfred Moessner in Volumes 19 and 20 of Scripta Mathematica, but it has proven difficult to find an archived copy to consult directly.


Sources