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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 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 languag ...
. 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, 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 exampl ...
''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 of ''b''" or, equivalently, "''b'' being an integer multiple of ''a''". This terminology is also used with units of measurement (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 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British English spelling differences#-re, -er, ...
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 pre ...
.. Section 4.3: ''Decimal multiples and submultiples of SI units: SI prefixes''. As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.


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 *
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 ...
* Decimal and SI prefix *
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 ...


References

{{DEFAULTSORT:Multiple (Mathematics) Arithmetic Multiplication Integers