PERFECT NUMBERS AND SOME ODDITIES 95
Multiplication by the numbers 8 to 13 gives related results:

142857 X 8=1142856

9=1285713
10=1428570
11 =1571427
12 =1714284
13=1857141

In each of these results there is, of course, an extra digit. Five
of the six original digits appear in each result; the other two
digits in each result, if added together, supply the missing
sixth original digit.

12. It has been contended that the numbers 142857 and
285714 are the only numbers (lower than one million) which
can be multiplied merely by removing the first digit and
adding it on again at the end. This statement is not
strictly correct. There are two exceptions, one of them
fundamental.

(a) Every number consisting of x digits can be shown as a
number of x+1 digits, where the first digit is a nought. The
number 1234 is really the same as o1234. The nought in the
position shown is always omitted in practice but it is always
understood to be there. Its inclusion means precisely the
same as its omission (that is, that there are no units of the
fourth power of 10 in the composition of the number). The
number 01234 is obviously multiplied by 10 when the nought
is taken from the first position and added on in the end
position.

(b) Apart from the foregoing, it may be true that the two
numbers quoted are the only ones which can be multiplied
by a whole number by moving the first digit to the other end.
It is certainly not true if we include multiplication by quite
simple fractions.

Thus:
153846 X 2=538461
This new number (153846) is exactly twice 76923. If this be

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