120 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

The first integers in the successiv groups will be the various
multiples of ,

«*, — 3m, — 2m, — m, 0, m, 2m, 3m, - - -, if m is positiv,
e+, 3m,2m,m,0, —m, — 2m, — 3m, - - -, if m is negativ.

If gm is any of these multiples, the group of which it is the
first integer will be

qm,gm+1,gm+2,gm+3,--',qm—l—(]m[ _1)

These integers are, in each case, evidently congruent re-
spectivly to 0, 1, 2, 3, - -+, |m| — 1. § 199.
Thus, if 4 is the modulus, the integers

_41 —'3; _2v _1)0’ 1y2’3y415)6y7
are congruent respectivly to
Ry a1 2.3.0.1,2.3

290. Th. If we replace each integer in the natural series by
that integer of the set 0, 1, 2, 3, -+, |m| — 1 to which it is
congruent with respect to the modulus m, this set of numbers will
be repeated in this order periodically.

291. Def. On this account the principle of § 290 is called
the principle of the periodicity of the integers with respect to
a given modulus.

292. Th. If m is any modulus and we have given any set of
|m| successiv integers, then there exists some positiv or zero
integer r, numerically less than m, such that the given set of
integers are congruent respectivly to the numbers r,r 4+ 1, r + 2,
e sy £ e bl =) e ] — 1)

Let a be the first integer in the given set.

Then 7 is the positiv or zero remainder got by dividing a
by m, using ordinary division. §§ 202, 78.

The successors of a in the given set are

a‘+1)a+2) "'ya+(lml —2),a—{—(|m| _1)