fractions with a common denominator. For instance,
+ = and – = .
It is possible to rewrite the terms of an arbitrary
collection of fractions so that they all share a common
denominator. For instance, the rewriting and as
and , respectively, shows that the two fractions have
a common denominator 15. In fact, any
COMMON
MULTIPLE
of 3 and 5 serves as a common denominator
of and . For instance, we have = and = ,
and = and = .
The
LEAST COMMON MULTIPLE
of the denominators
of a collection of fractions is called the least common
denominator of the fractions. For example, the least
common denominator of and is 15, and the least
common denominator of , , and is 24. One adds
and subtracts arbitrary fractions by rewriting those
fractions in terms of a common denominator.
common factor (common divisor) A number that
divides two or more integers exactly is called a com-
mon factor of those integers. For example, the numbers
20, 30 and 50 have 2 as a common factor, as well as 1,
5, and 10 as common factors. It is always the case that
the largest common factor a set of integers possesses is
a multiple of any other common factor. In our example,
10 is a multiple of each of 1, 2, and 5. The value 1 is
always a common factor of any set of integers.
The
FUNDAMENTAL THEOREM OF ARITHMETIC
shows
that any number can be uniquely expressed as a product
of
PRIME
factors. Any common factor of two or more
integers is a product of primes common to all those inte-
gers, and the largest common factor is the product of all
the primes in common, with repetition permissible. (This
explains why the largest common factor is a multiple of
any other common factor.) If the integers have no primes
in common, then their largest common factor is one.
See also
GREATEST COMMON DIVISOR
;
RELATIVELY
PRIME
.
common multiple A number that is a multiple of two
or more other numbers is called a common multiple of
those numbers. For example, 60 is a common multiple
of 5, 6, and 10. The lowest number that is a common
multiple of a given set of numbers is called their
LEAST
COMMON MULTIPLE
. In our example, 30 is the least com-
mon multiple of 5, 6, and 10. Every common multiple is
a multiple of the least common multiple.
The
FUNDAMENTAL THEOREM OF ARITHMETIC
shows that any number can be uniquely expressed as a
product of
PRIME
factors. Any common multiple of two
or more integers is the product of, at the very least, all
the primes that appear in the factorizations of the given
integers, with the necessary repetitions, with perhaps
additional factors. With no additional factors present,
one obtains the least common multiple.
commutative property A
BINARY OPERATION
is said
to be commutative if it is independent of the order of
the terms to which it is applied. More precisely, an
operation * is commutative if:
a*b = b*a
for all values of aand b. For example, in ordinary
arithmetic, the operations of addition and multiplica-
tion are commutative, but subtraction and division are
not. For instance, 2 + 3 and 3 + 2 are equal in value,
but 2 – 3 and 3 – 2 are not.
If an operation is both commutative and
ASSOCIA
-
TIVE
, then all products of the same set of elements are
equal. For example, the quantity a*(b*c) equals
(a*c)*b and b*(c*a).In this case, one is permitted to
simply write a*b*c, with terms in any order, without
concern for confusion.
In
SET THEORY
, the union and intersection of two
sets are commutative operations. In
VECTOR
analysis,
the addition and
DOT PRODUCT
of two vectors are com-
mutative operations, but the
CROSS PRODUCT
operation
is not. The multiplication of one
MATRIX
with another
is not, in general, commutative.
Geometric operations generally are not commuta-
tive. For example, a reflection followed by a rotation
does not usually produce the same result as performing
the rotation first and then applying the reflection. One
could also say that the operations of putting on one’s
shoes and one’s socks are not commutative.
A
GROUP
is called commutative, or Abelian, if the
operation of the group is commutative.
See also N
IELS
H
ENRIK
A
BEL
;
DISTRIBUTIVE PROP
-
ERTY
;
RING
.
5
6
3
4
3
8
2
5
1
3
18
45
2
5
15
45
1
3
12
30
2
5
10
30
1
3
2
5
1
3
6
15
5
15
2
5
1
3
2
12
3
12
5
12
8
12
3
12
5
12
commutative property 83