| Casting Out Nines |
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If ''x'' and ''x''' (respectively, ''y'' and ''y''') have the same remainder Modulo 9, then so do ''x'' + ''y'' and ''x''' + ''y''', ''x'' − ''y'' and ''x''' − ''y''' and ''x'' × ''y'' and ''x''' × ''y'''. The trick is that there exists a simple way to compute the remainder of a number by 9, based on the following check: the sum of the Digit s of the Decimal writing of an integer has the same remainder modulo 9 as this integer. One therefore can sum all digits in the original number to obtain another number, and so on repeatedly until one gets a 1-digit number, which is necessarily equal to the original number. The method takes its name from the fact that nines may be ignored when doing these summations, since they are equal to 0 modulo 9. You can also return to zero whenever you reach a sum which is 9 or a multiple of nine like 18. Note that 18 (add digits) = 9, and so on for 27, 36, 45, etc. . This description sounds more complicated than the method really is. Let us for instance consider 19786901 × 8098678443. The correct result is 160247748582475143, but we are likely to make a mistake if we compute it by hand unless we show great attention. Let us now apply the above summing process: 19786901 ⇒ 41 ⇒ 5 8098678443 ⇒ 57 ⇒ 12 ⇒ 3 160247748582475143 ⇒ 78 ⇒ 15 ⇒ 6 5 × 3 = 15 ⇒ 6 indeed. If the results had differed, we would have had to conclude that we had made a mistake. However, since some errors (for example, a shortfall of some multiple of nine when adding) will not be reflected in the digital roots, a correct result after casting out nines does not guarantee that our result is correct — the technique catches most, but not all, random errors. Shortcut :The numerical example above can be solved even more quickly by casting nines out of the multiplier, multiplicand, and product. We can visually inspect each one for combinations of numerals that sum to nine, and cast them out. :By inspection, we see that the number 19786901 has two 9's. Cast them out (note strikethroughs): 1 :Similarly, we can cast nines out from 8098678443. (Note that there is not only one way to do it.) We cast out the 9, as well as the 3 and 6, as well as the 0: 8 :Lastly, cast nines out from 160247748582475143. We cast out these numerals by inspection: 1 and 8; 6 and 3; 0, 2 and 7; 4 and 5; 7 and 2; 4 and 5; 8 and 1: :Once again, 5 × 3 = 15 ⇒ 6. :Indeed, it is this shortcut which gives "casting out nines" its name. HISTORY Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was later employed by twelfth-century Hindu mathematicians. REFERENCES
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