Anomalous Cancellation

15̸717\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, ABBC=AC\frac{\overline{AB}}{\overline{BC}} = \frac{A}{C}. For how many fractions would this method of simplification work?

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

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