Anomalous Cancellation

The simplification of a Fraction $a/b$ 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: $64/16=4/1=4$, $98/49=8/4=2$, $95/19=5/1=5$, and $65/26=5/2$ (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 $b-1$ is Prime, this type of solution is the only one. For base 4, for example, the only solution is $32_4/13_4=2_4$. Boas gives a table of solutions for $b\leq 39$. 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

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.