T P A T A

i
a
g
;

i g A

 

e A

e

e s o A Aot~ e, IS T SN T~
o S g

34 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Ex. 2.

 

 

137 = 129 4 2-4
120 + — 13.4 + — 3.4 X (— 5)
120 + — 13-4 4+ — 114 X (= 5) +4-4 X (= 5) X 2

93. If in the theorem of § 92 the divisors 7y, 7o, 73, +++ are
all equal to 0, the remainders aq, a1, as, --- are respectivly
equal to qo, ¢1, @2, -

We will now assume that none of the divisors is zero but
suppose them all equal. The theorem of § 92 then gives us
as a special case the following theorem.

94. Th. If a is an integer, v a positiv or negativ integer, and
G, @2, qs, *** any series of integers, if qo = a and ay_,, where
k = 1, is the remainder got by dividing qi_, by r using the quo-
tient qx, then

Il

a = Qo

k—1
> {alrl} + qur*
=0

= a0+ arr + agr? + - + @yl - gt
ror, ik =1,

k—1 l k 0 l 1
> {azn <rm>} Fallt) =% {GZH <rm>} ST
=0 m=0 m=0 =0 m=0 m=0

0 1
= ao Il () + ¢ II (rn) = aoro + qrors = a0 + qiny
m=0 m=0