20 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

5 = g X 2; the integer 3 is not divisible by the integer 0; no
integer, not even 0, is divisible by 0.(%)

This idea of “‘being divisible by " is a relation that may or
may not exist between two given integers, just as ‘‘being
father to” is a relation between men.

55. Def. If a is divisible by b, then b is said to be a divisor
of, or factor of, . Thus 3 is a divisor of 9.

The relation of “being a divisor of” between integers is
the converse of the relation of ‘“being divisible by,” just as

““being son of”’ is the converse of “being father to’ amon
men.

56. For the relations of ‘‘being divisible by and ‘‘being
a divisor of ’ we will use the signs >> and < respectivly.

Thus 8 22,3 < 15.

For the quotient of ¢ and b we will use the symbol a : b,
this symbol standing for the result of an operation on @ and b
called division.

Thus 8 : 4 = 2,

57. By means of these definitions the reader can probably
easily prove the following theorems,(?) which may be easily
verified by using particular numbers.

58. Th. [Every integer except zero is divisible by itself, the
quotient being unaty.
In symbols: If a —= 0, then a >> aand a :a = 1.

() A word of warning may here be worth while. According to the
definition of quotiéent of integers given above there is no quotient of 5 and
2 nor of 3 and 0. But according to the definition of quotient of rational
numbers, the next kind of numbers after integers, there may be (in fact
are) quotients of the rational numbers named 5 and 2 and of those named
3 and 0, these quotients being the fractions five halves and infinity.

In trigonometry the tangent of 90° is the number infinity, altho many
authors say there is no such thing as the tangent of 90°.

(%) Proofs of these theorems may also be found in the author’s Elements
of the Theory of Integers, Chapter VI.