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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a multiple is the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of any quantity and an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. In other words, for the quantities ''a'' and ''b'', it can be said that ''b'' is a multiple of ''a'' if ''b'' = ''na'' for some integer ''n'', which is called the multiplier. If ''a'' is not
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, this is equivalent to saying that b/a is an integer. When ''a'' and ''b'' are both integers, and ''b'' is a multiple of ''a'', then ''a'' is called a
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of ''b''. One says also that ''a'' divides ''b''. If ''a'' and ''b'' are not integers, mathematicians prefer generally to use integer multiple instead of ''multiple'', for clarification. In fact, ''multiple'' is used for other kinds of product; for example, a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
''p'' is a multiple of another polynomial ''q'' if there exists third polynomial ''r'' such that ''p'' = ''qr''. In some texts, "''a'' is a submultiple of ''b''" has the meaning of "''a'' being a
unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
of ''b''" or, equivalently, "''b'' being an integer multiple of ''a''". This terminology is also used with
units of measurement A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...
(for example by the BIPM and NIST), where a ''submultiple'' of a main unit is a unit, named by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 103. For example, a millimetre is the 1000-fold submultiple of a
metre The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
.. Section 4.3: ''Decimal multiples and submultiples of SI units: SI prefixes''. As another example, one
inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth") ...
may be considered as a 12-fold submultiple of a
foot The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg made ...
, or a 36-fold submultiple of a
yard The yard (symbol: yd) is an English unit of length in both the British imperial and US customary systems of measurement equalling 3  feet or 36 inches. Since 1959 it has been by international agreement standardized as exactly ...
.


Examples

14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such ''integers'' for 3 and −6. Each of the products listed below, and in particular, the products for 3 and −6, is the ''only'' way that the relevant number can be written as a product of 7 and another real number: : 14 = 7 \times 2; : 49 = 7 \times 7; : -21 = 7 \times (-3); : 0 = 7 \times 0; : 3 = 7 \times (3/7), \quad 3/7 is not an integer; : -6 = 7 \times (-6/7), \quad -6/7 is not an integer.


Properties

* 0 is a multiple of every number (0=0\cdot b). * The product of any integer n and any integer is a multiple of n. In particular, n, which is equal to n \times 1, is a multiple of n (every integer is a multiple of itself), since 1 is an integer. * If a and b are multiples of x, then a + b and a - b are also multiples of x.


See also

*
Unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
*
Ideal (ring theory) In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pr ...
*
Decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
and
SI prefix The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
*
Multiplier (linguistics) In linguistics, more precisely in traditional grammar, a multiplier is a word that counts how many times its object should be multiplied, such as ''single'' or ''double''. They are contrasted with distributive numbers. In English, this part of ...


References

{{DEFAULTSORT:Multiple (Mathematics) Arithmetic Multiplication Integers