200 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

then the final result s,

if m = 2, a number of two digits both equal to r — 1; but,

if m > 2, 1t1s a number of m + 1 digits,
of which the first two are 1, 0, the last twor — 2 and r — 1,
and the others, if there are any, are equal tor — 1;

again, if v > 2, the final result is v — 1 times a number of m
digits of which the first and last are I's and the others, if there
are any, are 2's; and in every case the final result is r* — 1 times.
a number of m — 1 digits, all of which are I’s.

N SR R A -

T e e e~ . e .

 

The rule given in § 478 for numbers written in the decimal
system and especially the rule for three figure numbers is

i evidently a consequence of the rule just given.

!" Examples.

i Radix 2. 1010 11000

i

\; 0011 01001

i 0111 01111

b 1110 11110

5:1; 10101 101101

i

i Radix 3. 221 200

i §3- 002

4 ; 022 121

] 220 121

fi! 1012 1012

b

i Radix 4. 3211 2221

i) e 1222
1332 0333
2331 3330

10323 10323