CHAPTER III

CONGRUENCE

154. Before introducing the main subject of this chapter,
that of congruence, let me review briefly certain other ideas
of arithmetic ().

COMPOSIT PRIME

155. Def. An integer is said to be composit, if it has some
positiv factor that is different from its numerical value and
from unity. All other integers are called prime. That is, a
prime integer is an integer that has no positiv factor that is
different from its numerical value and from unity.

Thus, since 0 has the factor 2, which is different from 0
and from 1, 0 is composit. Similarly 4, 6, 8, 9, 10, 12, 14,
15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28 are composit, as are
also their opposits — 4, — 6, — 8, ---; whereas |1, 2, 3,5,7, 11,
13, 17, 19, 23, 29 and their opposits are prime (?).

156. Th. If a is prime and is divisible by b, where |b| —= 1,
then |a| = |b]|.

PRIME TO

157. Def. a ]| b, which is read ‘“a is prime to b,”” means
that ¢ and b have no positiv common factor except unity.
For example, 4 ]| 7.

158. Th. ‘@&lLi>b il a
159. Th. allbD —0allb—~all -0~ —all —0b

(1) A fuller treatment will be found in the author’s Elements of the
Theory of Integers.

(2) According to some authors 1 is not prime. Neither is it composit!
In olden days they used even to say that one is not a number (unus non
est numerus).

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