Anomalous cancellation
An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. An attempt is made to reduce a fraction by cancelling individual digits in the numerator and denominator. This is not a legitimate operation, and does not in general give a correct answer, but in some rare cases the result is numerically the same as if a correct procedure had been applied. The trivial cases of cancelling trailing zeros or where all of the digits are equal are ignored. Examples of anomalous cancellations which still produce the correct result  include (these and their inverses are all the cases in base 10 with the fraction different from 1 and with two digits): The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32/13 = 2/1 and its inverse are the only solutions in base 4 with two digits. The anomalous cancellation happens also with more digits, e.g. 165/462 = 15/42 and those with different numbers of digits (98/392 = 8/32). Elementary properties. When the base is prime no two-digit solutions exist. This can be proved by contradiction: suppose a solution exists, and without loss  of generality we can say that this solution is formula_4 where the line indicates digit concatenation. Thus we have formula_5 But formula_6 as they are digits in base formula_7 yet formula_8 which means that formula_9 so therefore the right hand side is zero which means the left hand side must also be zero, i.e. formula_10, a contradiction. Another property is that the numbers of solutions in a base formula_11 is odd if and only if formula_11 is an even square. This can be proved similarly to the above: suppose that we have a solution formula_4 Then doing the same manipulation  we get formula_14 Suppose that formula_15. Then note that formula_16 is also a solution to the equation. This almost sets up an involution from the set of solutions to itself, but a problem arises when formula_17. In this case, we can substitute in to get formula_18 so this only has solutions when formula_11 is a square. Let formula_20. Square rooting and rearranging yields formula_21. Since the greatest common divisor of formula_22 is one, we know that formula_23. Noting that formula_24, this has precisely the solutions formula_25 i.e. it has an odd number of solutions when formula_26 is an even square . The converse of the statement may be proved by noting that these solutions all satisfy the initial requirements.