Anomalous Cancellation/Examples/143 185 over 17 018 560

Theorem
The fraction $\dfrac {143 \, 185} {17 \, 018 \, 560}$ exhibits the phenomenon of anomalous cancellation:


 * $\dfrac {143 \, 185} {17 \, 018 \, 560} = \dfrac {1435} {170 \, 560}$

as can be seen by deleting the $18$ from both numerator and denominator.

This is part of a longer pattern:
 * $\dfrac {1435} {170 \, 560} = \dfrac {143 \, 185} {17 \, 018 \, 560} = \dfrac {14 \, 318 \, 185} {1 \, 701 \, 818 \, 560} = \cdots$

Proof
Let $q = \dfrac r s = \dfrac {1431818 \cdots 185} {1701818 \cdots 18560}$.

Let the number of digits in $r$ be $n + 2$.

Then the number of digits in $s$ is $n + 4$.

By inspection, it is seen that $n$ is even.

Then it is seen that:

and similarly:

Hence: