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© Copyright 1999, Jim Loy
I received email (in response to Casting Out Nines) asking if casting out elevens is better than casting out nines, as a means of checking arithmetic. I replied that I had never heard of casting out elevens. Well, it is easy to deduce what casting out elevens might be. When you cast out nines, the final one-digit number that you get is congruent to the answer, modulo 9. See Modular Arithmetic. If we can find a number that is congruent to the answer, mod 11, that would work about as well. An example:
873 4
x257 4
------
6111
4365
1746
------
224361 5
the 4, 4, and 5, to the right, are congruent to the various numbers, mod 11: 873 is congruent to 4 (mod 11), 257 is congruent to 4 (mod 11), and 224361 is congruent to 5 (mod 11). Well, 4x4=16 which is congruent to 5 (mod 11). So this checks out.
Let's see if we can actually "cast out" any elevens. The most obvious eleven is the 22 at the left end of our answer. We can cross that out, giving 4361, without changing the 5 (mod 11), as we are just subtracting 220000, which is divisible by 11. Elsewhere, we can subtract 770 from 873 and get 103, then we can subtract 99, to get 4 (mod 11). Instead of subtracting 99, we can just add the 1 and 3 and get 4 (mod 11). This works because 100 is congruent to 1 (mod 11), as is 10000, and 1000000 (any even number of zeros). A person can become adept at this.
So we can actually cast out elevens. It is slightly more accurate than casting out nines (we are slightly more likely to catch an error). But casting out elevens is more difficult.
See Divisibility Tests.