If it be multiplied by multiples of 3, beyond 27, this peculiarity is continued, except that the extreme figures taken together represent the multiple of 3 that is used as a multiplier. Thus—
- 37 × 30 = 1110, extreme figures, 10
- 37 × 33 = 1221 " " 11
- 37 × 36 = 1332 " " 12
The number 73 (which is 37 inverted) multiplied by each of the numbers of arithmetical progression 3, 6, 9, 12, 15, etc., produces products terminating (unit’s place) by one of the ten different figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These figures will be found in the reverse order to that of the progression, 73 × 3 produces 9, by 6 produces 8, and 9 produces 7, and so on.
Another number which falls under some mysterious law of series is 142,857, which, multiplied by 1, 2, 3, 4, 5, or 6 gives the same figures in the same order, beginning differently; but if multiplied by 7, gives all 9’s.
| 142,857 | multiplied by | 1 | = 142,857 |
| " | " | 2 | = 285,714 |
| " | " | 3 | = 428,571 |
| " | " | 4 | = 571,428 |
| " | " | 5 | = 714,285 |
| " | " | 6 | = 857,142 |
| " | " | 7 | = 999,999 |
Multiplied by 8, it gives 1,142,856, the first figure added to the last makes the original number—142,857.
The vulgar fraction 1⁄7 = ·142,857.
The following number, 526315789473684210, if multiplied as above, will, in the product, present the same peculiarities, as also will the number 3448275862068965517241379310.
| The multiplication of | 987654321 | by 45 | = 444444444445 |
| Do. | 123456789 | " 45 | = 5555555505 |
| Do. | 987654321 | " 54 | = 53333333334 |
| Do. | 123456789 | " 54 | = 6666666606 |
Taking the same multiplicand and multiplying by 27 (half 54) the product is 26,666,666,667, all 6’s except the extremes, which read the original multiplier (27). If 72 be used as a multiplier, a similar series of progression is produced.