How It Works
The method works because the original numbers are 'decimal' (base 10), the modulus is chosen to differ by 1, and casting out is equivalent to taking a digit sum. In general any two 'large' integers, x and y, expressed in any smaller modulus as x' and y' (for example, modulo 7) will always have the same sum, difference or product as their originals. But this property is also preserved for the 'digit sum' where the 'base' and the 'modulus' differ by 1.
To see this take an example: Both 900 and 630 are exactly divisible by 9 and have the same digit sum - '63' changes into '90' by repeated addition of '09' and the change in the second digit always offsets the change in the first ('63' to '72' to '81' to '90'). For two 'decimal' numbers not generally congruent modulo 9 (like '914' and '673') the picture remains the same; we can consider their congruences pairwise ('900' with '630') plus ('09' with '36') plus ('5' with '7'). Thus the digit sum of any number and its congruence modulo 9 are always fixed in base 10.
If a calculation was correct before casting out, casting out on both sides will preserve correctness. However, it is possible that two previously unequal integers will be identical modulo 9 (on average, a ninth of the time).
One should note that the operation does not work on fractions, since a given fractional number does not have a unique representation.
Read more about this topic: Casting Out Nines
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