The simplification of a Fraction which gives a correct answer by ``canceling'' Digits of and . There are only four such cases for Numerator and Denominators of two Digits in base 10: , , , and (Boas 1979).
The concept of anomalous cancellation can be extended to arbitrary bases. Prime bases have no solutions, but there is a solution corresponding to each Proper Divisor of a Composite . When is Prime, this type of solution is the only one. For base 4, for example, the only solution is . Boas gives a table of solutions for . The number of solutions is Even unless is an Even Square.
4 | 1 | 26 | 4 |
6 | 2 | 27 | 6 |
8 | 2 | 28 | 10 |
9 | 2 | 30 | 6 |
10 | 4 | 32 | 4 |
12 | 4 | 34 | 6 |
14 | 2 | 35 | 6 |
15 | 6 | 36 | 21 |
16 | 7 | 38 | 2 |
18 | 4 | 39 | 6 |
20 | 4 | ||
21 | 10 | ||
22 | 6 | ||
24 | 6 |
See also Fraction, Printer's Errors, Reduced Fraction
References
Boas, R. P. ``Anomalous Cancellation.'' Ch. 6 in Mathematical Plums (Ed. R. Honsberger).
Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86-87, 1988.
© 1996-9 Eric W. Weisstein