188 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

474. Innumerable other rules may be derived from §§ 98,
102 by giving 7 various values, positiv and negativ.

These other rules, however, are not so simple in statement
as the three given except the rule obtaind from § 102 by taking
r equal to 10 (in any scale of notation). This rule is prac-
tically the same as the ordinary rule for multiplication.

A QUEER PROPERTY OF NUMBERS (!)

475. A boy asks you to write the number 12,345,679.
You do so and he asks, ‘“Which figure looks the worst?”
Suppose you answer, ‘‘4."”
“Multiply by 36,” says he.
This you do as follows:
12345679
36

74074074
37037037

A

““Can you write it any better now?" he asks triumphantly.
Here is the explanation.

Let a = 12345679
Then g X9=1111111%§
Hence aX18=e X (9X2) =(aX 9 X 2=222222223
g 2 =a X {(9X3)=10X9 X 3=2333333313
aX36=aX(9X4) (a><9)><4»—-14—1444444
aX81=a><(9><9)—(a><9)><9=999999999

476. That 12345679 X 9 = 111 111 111 (radix ten) is a
special case of the following theorem:
Th. If the radix r is greater than 3,

I:rf)’e'r’“’”“2 + (r — 1)](7 — 1)

(1) See E. Lucas, Récréations Mathématiques, Vol. 1V, pp. 232, 3;
E. Lucas, Théorie des Nombres, Vol. I, p. 8; Wm. F. White, 1. c., p. 18.