142 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

since the numerical value of the sum of any number of num-
bers, none of which is negativ or none of which is positiv, is
equal to the sum of their numerical values.

ké) (tr)

Thus the new number obtaind by the method of § 341 is
numerically smaller than the given number.

But this new number, since it is obtaind by adding numbers
which are all positiv or zero, or all negativ or zero, will, if the
adding is done in the usual way, itself have its digits either
all positiv or zero or all negativ or zero.

If this new number is not a one-period number, the same
process may be applied to it and we will obtain a number
which is numerically still smaller.

Hence, since there are only a finite number of numbers
between a given positiv number |a| and zero, this process
must finally give a one-period number.

344. From the theorems of §§ 334-343 we may get special
theorems by putting m =1, 2, 3, -~

We will mention a few of these.

Th. If the modulus is r — 1, any number may be subtracted
from any digit, if the same number is added to any other digit.

For example, 72983 = 52985 (mod r — 1)

345. Th. As a consequence, any two numbers which have the
same digits, regardless of order, and disregarding zero digits, are
congruent, if the modulus is r — 1.

For example, 500 =50 =5 (modr — 1)

2700 = 2070 = 720 = 27 (mod r — 1)

346. Th. ea=a,+ a1+ -+ + a2+ a1+ ao(modr — 1)

In words: Any number is congruent to the sum of its digits,
if the modulus is the immediate predecessor of the radix.

This theorem is a special case of either § 341 or § 335. It
may also be proved from the formula

@ = 0ut" + Qa1+ o+ a2’ + arr + a0
using the fact that » = 1 (mod » — 1).

Therfor |a| > yorla|l > [po+pr+pe+ -+ + pil