94 THE FASCINATION OF NUMBERS

10. It is possible to find certain numbers of eight different
digits which, when multiplied by the remaining ninth digit,
will give results containing all nine digits:

51249876 X 3=153749628
16583742 X 9=149253678
32547891 X 6=195287346
and also some variants:
(a) where a zero is introduced in the result:
675412398 X 2 =1350824796
(b) using a two-digit number as multiplier:
8745231 X 96 =839542176
(¢) using only some of the digits:

8 X473=3784
15 X93=1395
35 X41=1435

(d) where the same number can be obtained by the multi-
plication of two different sets of numbers which are never-
theless composed of the same digits:

12 X483 =42 X 138=5796
18 X297 =27 X198 =5346

11. The number 142857 has the remarkable property that
if it is multiplied by any number from 1 to 6, the results will
always consist of the same digits in the same cyclic order, but
commencing at different points.

142857 X 1=142857

2=285714
3=428571
4=571428
5="714285
6=3857142

Multiplication by 7, however, gives a different result:

142857 X 7=999999