15 × 9 = 45, and 4 + 5 = 9
16 x 9 = 54, and„ 5 + 4 = 9
17 × 9 = 63, and„ 6 + 3 = 9
18 × 9 = 72, and„ 7 + 2 = 9
19 × 9 = 81, and„ 8 + 1 = 9
10 × 9 = 90, and„ 9 + 0 = 9
It will be noticed that 9 × 11 makes 99, the sum of the digits of which is 18 and not 9, but the sum of the digits 1 x 8 equals 9.
9 × 12 = 108, and 1 + 0 + 8 = 9
9 × 13 = 117, and„ 1 + 1 + 7 = 9
9 × 14 = 126, and„ 1 + 2 + 6 = 9
And so on to any extent.
M. de Maivan discovered another singular property of the same number. If the order of the digits expressing a number be changed, and this number be subtracted from the former, the remainder will be 9 or a multiple of 9, and, being a multiple, the sum of its digits will be 9.
For instance, take the number 21, reverse the digits, and you have 12; subtract 12 from 21, and the remainder is 9. Take 63, reverse the digits, and subtract 36 from 63; you have 27, a multiple of 9, and 2 + 7 = 9. Once more, the number 13 is the reverse of 31; the difference between these numbers is 18, or twice 9.
Again, the same property found in two numbers thus changed, is discovered in the same numbers raised to any power.
Take 21 and 12 again. The square of 21 is 441, and the square of 12 is 144; subtract 144 from 441, and the remainder is 297, a multiple of 9; besides, the digits expressing these powers added together give 9. The cube of 21 is 9261, and that of 12 is 1728; their difference is 7533, also a multiple of 9.
