CONGRUENCE 133

312. Def. If, now, starting at the units digit a number is
divided into periods of m figures each, except the last period,
which may have less than m figures when m > 1, we will call
these periods the m-figure periods of the number.

Thus the 2-figure periods of the number 364725870 are 3,
64, 72, 58, and 70; its 3-figure periods are 364, 725, and 870.

We will also use the term period to stand for the number
represented by the period.

313. Th. If po, p1, P2, s, =+ *, D1 are the successiv m-figure
periods of the number a, py containing the units digit, then

po=ao+ arr + axr* + -+ + amar™!

P1 = Qmn + QGmi1? + Qi + 2 + Qg ™!

P2 = Qom + Gompa? + Aomgo?? + o0 + Cgpar™ !

D1 = Cim + Gima? + Qimye?? + 200 F @p1ym1r™ !
and @ = po+ pur™ + por®™ 4 pard™ 4 oo 4 pyrim

For example, 267,325,791 = 791 + 325¢% 4 267/°
We may also represent the period numbers po, p1, P2, * * *, P2
and the number a as follows:

m—1

po = kg() (axr*)

2m—1

P = 2 (awrF™)
k=m
3m—1

pr = 2 (awr*?m)
k=2m
(I4+1)ym—1

pr= 2 ('™
k=1Ilm

l
a = 2 (pur*™)
k=0