The theory of numbers, by Robert D. Carmichael ...

ELEMENTARY PROPERTIES OF INTEGERS

9

II. If c is a divisor of both a and b, then c is a divisor of the
sum of a and b.
From the hypothesis of the theorem it follows that integers
a and A exist such that
a=ca, b-=c.
Adding, we have
a+b =ca+c3 = c(a+3) =c6,
where 3 is an integer. Hence, c is a divisor of a+b.
III. If c is a divisor of both a and b, then c is a divisor of the
difference of a and b.
The proof is analogous to that of the preceding theorem.
DEFINITIONS. If a and b are both divisible by c, then c
is said to be a common divisor or a common factor of a and b.
Every two integers have the common factor i. The greatest
integer which divides both a aned b is called the greatest common
divisor of a and b. More generally, we define in a similar way
a common divisor and the greatest common divisor of n integers
ai, a2,...* an.
DEFINITIONS. If an integer a is a multiple of each of two
or more integers it is called a common multiple of these integers.
The product of any set of integers is a common multiple of the
set. The least integer which is a multiple of each of two or
more integers is called their least common multiple.
It is evident that the integer I is a divisor of every integer
and that it is the only integer which has this property. It is
called the unit.
DEFINITION. Two or more integers which have no common
factor except I are said to be prime to each other or to be relatively prime.
DEFINITION. If a set of integers is such that no two of
them have a common divisor besides i they are said to be prime
each to each.
EXERCISES
i. Prove that n3-n is divisible by 6 for every positive integer n.
'2. If the product of four consecutive integers is increased by i the result
is a square number.
3. Show that 24n+2+I has a factor different from itself and I when n is a
positive integer.