CONGRUENCE 119

PERIODICITY WITH RESPECT TO A MODULUS

285. Th. With respect to any modulus m every integer 1s
congruent to one of the integers 0, 1, 2, 3, « -+, |m| — 1.

This follows immediately from §§ 202, 80, since every in-
teger when divided by the modulus leaves a positiv or zero
remainder numerically less than the divisor, if the quotient is
exact or is a lower approximate one.

286. Th. No two of the numbers 0, 1, 2, 3, «++, |m| — 1
are congruent to each other with respect to the modulus m.
For let £ and ! be two of these numbers.

Then k—= I, 0=k < |m|
and lm| >1=0
By subtraction — |m| <k —1 < |m|
Hence |k — 1| < |m] § 46.
Since, now, k—= land |k — I| < |m|, k—= 1 (mod m)

§ 181.

287. Th. With respect to any modulus m no integer is con-
gruent to more than one of the integers

G 1.2 3 v, i}

For, if it were congruent to two of these numbers, these
two integers would be congruent to each other, which would
contradict § 286.

288. Th. With respect to any modulus m every integer is con-
gruent to one, and only one, of the integers

0,1,2,3, ¢+, |m| — 1
This follows immediately from §§ 285, 287.

289. Th. If m is any given modulus, the natural series of
integers may be divided into successiv groups of |m| each in
number, the groups beginning with the multiples of m and the
integers in each group, taken in order, being congruent to 0, 1,
2,3, +++, |m| — 1 respectivly.