Anomalous Cancellation

The simplification of a fraction which gives a correct answer by "canceling" digits of a and b. 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 b. 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 b is an even square.

b	N	b	N
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

Fraction, Printer's Errors, Reduced Fraction

Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.

Moessner, A. Scripta Math. 19.

Moessner, A. Scripta Math. 20.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86-87, 1988.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 26-27, 1986.