If any row of two or more figures be reversed and subtracted
from itself, the figures composing the remainder will, when added,
be a multiple of 9, and if added together continually will result in 9.
Example: 7362
2637
----
4725 = 18 = 9
Tell a person to write a row of figures, then to add them together,
and to subtract the total from the row first written, then to cross out
any one of the figures in the answer, and to add the remaining
figures in the answer together, omitting the figure crossed out; if the
total be now told, it is easy to discover the figure crossed out.
Example: 4367256 = 33
33
-------
4367223 = 27
It should be observed that the figures of the answer to the
subtraction when added together equal 27--a multiple of 9; this,
of course, is always the case. Now, suppose that 7 was the figure
crossed out, then the sum of the figures in the answer (omitting
7) would be 20; this number being told by the person, it is easily
seen that 7 must have been crossed out, as that figure is required
to complete the multiple 27. If after the figure has been crossed
out, the remaining figures total a multiple of 9, it is evident that
either a cipher or a 9 must have been the figure erased.
Multiply the digits--omitting 8--by any multiple of 9, and the
product will consist of that multiple,
Example: 12345679 36 = 4 x 9
36
--------
444444444
If a figure with a number of ciphers attached to it be divided by
9, the quotient will be composed of that figure only repeated as many
times as there are ciphers in the dividend; with the same figure as
the remainder.
Example: 9)7000000
--------
777777-7