(1) 90 = 9 (2) 54 = 9 (3) 1458 = 18 = 9 (4) 135 = 9
Also, if these answers be multiplied by any number whatever, a similar result will be produced.
Example: 135 x 8 = 1080 = 9
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 |