# Anomalous Cancellation

Number Theory Level 5

$\large\dfrac{1\not{5}}{\not{5}7}\ne\dfrac{1}{7}$

When teaching children how to reduce fractions, many of them will make mistakes like the one shown above. However, there are a few situations in which making this error does yield the correct answer, that is, $$\frac{\overline{AB}}{\overline{BC}} = \frac{A}{C}$$. For how many fractions would this method of simplification work?

Note: $$\overline{AB}$$ simply denotes the decimal expansion of a two-digit number with $$A$$ and $$B$$ as the digits.

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