PERFECT NUMBERS AND SOME ODDITIES 93

digits in ascending order), if multiplied, gives the following
results:
Multiply by 2 246913578

iy 8 SI0NI0a8T

s o» 4 493827156

» w5 617283045

s » 6 740740734

» o 7 8064197523

»  » 8 987654312
In each case, except where the multiplier is a multiple of 3,
all the nine digits are repeated again in the result although
in no consistent order. When the multiplier is a multiple of g,
the first three digits of the result are repeated to give the next
three digits, whilst the same three digits minus the multi-
plier give the last three digits of the result. Similarly, related
results introducing the zero sign in the answer, can be ob-
tained by multiplying by some higher numbers.

8. If the number in the last paragraph is reversed, we
obtain the number 987654321. Multiplication of this number
gives the following pattern numbers:

Multiply by 2 1975308642
»  » 4 3950617284
»  » 5 4938271605
7 6913580247
»  » 8 7901234568
All the digits reappear with the addition of the zero sign in
the results.

9. If the original two numbers in the last two paragraphs
are written in the form of a subtraction, a new number, still
containing all the digits, will appear.

987654321
123456789
864197532

2 2»

This number is very nealzly the same as was obtained for the
multiplier 7 in paragraph 7 above.

R R RECO TR )

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