CONGRUENCE 125

successiv periods of m each, the periods beginning with the results
of multiplying the multiples of m by a, that is, beginning with
the multiples of am, and the products in each period, taken in
order, being respectivly congruent (mod m) to some arrangement
of the numbers 0, 1, 2, «++, m — 1, the first product in each
period being congruent to 0.

For the natural series of integers may be divided into suc-
cessiv periods of m each, the periods beginning with the
multiples of m and such that, if b, ¢, d, - - -, k are the numbers
in any period, these are congruent respectivly to 0, 1, 2, -- -,
m — 1. § 289.

Hence ba, ca, da, - -+, ka are congruent respectivly to

Oa, 1a, 2a, -+, (m — 1)a, § 212.
the first of which is congruent to 0 and the others respectivly
to some arrangement of the numbers

1,2, vos, ni—¥, § 298.

300. Th. Ifall m > 1, if any set of m successiv integers
are multiplied by a, the results taken in order will be respectivly
congruent (mod m) to some arrangement of the numbers 0, 1, 2,
«eo.m — 1, the result of multiplying the multiple of m in the set
being congruent to 0.

For any set of m successiv integers are congruent in order
to some arrangement of the integers

0,1,2, «vs,m=1 § 294.

Hence the results of multiplying the integers in the given
set by a will be respectivly congruent to the corresponding
arrangement of

Oa, 1a, 2a, +++, (m — 1)a, § 212.

the first of which is congruent to 0 and the others respectivly
to some arrangement of the numbers

1,2, «er, =8, § 298.