The Second Book of Mathematical Puzzles and Diversions

Digital Roots 49
divided by 9. This is not hard to prove, and perhaps an in-
formal statement of a proof will interest some readers.
Consider a four-digit number, say 4,135. This can be writ-
ten as sums of powers of 10 :
If 1 is subtracted from each power of 10, we can write
the same number like this :
(4 x 999) + (1 x 99) + (3 x 9) + (5 x 0) + 4 + 1 + 3 + 5
The expressions inside the. parentheses are all multiples
of 9. After casting them out, we are left with 4 + 1 + 3 + 5,
the digits of the original number.
In general, a number written with the digits abed can
be written :
(aX999) + (bX99) + (cX9) + (dXO) +a+b+c+d
Therefore a + b + c + d must be a remainder after cer-
tain multiples of 9 are cast out. This remainder of course
may be a number of more than one digit. If so, the same
procedure will show that the sum of its digits will give
another remainder after other multiples of 9 are cast out,
and we can continue until only one digit, the digital root,
remains. Such a procedure can be applied to any number,
no matter how large. The digital root, therefore, is the num-
ber that remains after the maximum number of 9's have
been cast out; that is, after the number is divided by 9.
Digital roots are often useful as negative checks in deter-
mining whether a very large number is a perfect square or
cube. All square numbers have digital roots of 1, 4, 7 or 9,
and the last digit of the number cannot be 2, 3, 7 or 8. A
cube may end with any digit, but its digital root must be 1,
8 or 9. Most curiously of all, an even perfect number (and
so far no odd perfect number has been found) must end in
6 or 28 and, with the exception of 6, the smallest perfect
number, have a digital root of 1.