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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.



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© 1996-9 Eric W. Weisstein
1999-05-25