The Second Book of Mathematical Puzzles and Diversions

Iligitnl Roots 45
Obtaining the digital root is simply the ancient process
of "casting out 9's." Before the development of computing
devices, the technique was widely used by accountants for
checking their results. Some modern electronic computers,
the International Business Machine NORC, for example, use
the technique as one of their built-in methods of self-check-
ing for accuracy. The method is based on the fact that if
whole numbers are added, subtracted, multiplied or evenly
divided, the answer will be congruent modulo 9 to the num-
ber obtained by adding, subtracting, multiplying or dividing
the digital roots of those same numbers.
For example, to check quickly a sum involving large num-
bers you obtain the digital roots of the numbers, add them,
reduce the answer to a root, then see if it corresponds to the
digital root of the answer you wish to test. If the roots fail
to match, you know that there is an error somewhere. If
they do match, there still may be an error, but the proba-
bility is fairly high that the computation is correct.
Let us see how all this applies to the telephone-number
trick. Scrambling the digits of the number cannot change
its digital root, so we have here a case in which a number
with a certain digital root is subtracted from a larger num-
ber with the same digital root. The result is certain to be a
number evenly divisible by 9. To see why this is so, think
of the larger number as consisting of a certain multiple of
9, to which is added a digital root (the remainder when the
number is divided by nine). The smaller number consists
of a smaller multiple of 9, to which is added the same digital
root. When the smaller number is subtracted from the larger,
the digital roots cancel out, leaving a multiple of 9.
(A multiple of 9) + a digital root
- (A multiple of 9) + the same digital root
(A multiple o f9)+7