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    • Source: Multiple (mathematics)
    • In mathematics, a multiple is the product of any quantity and an integer. 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


      {\displaystyle 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 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 p is a multiple of another polynomial q if there exists third polynomial r such that p = qr.


      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
      ×
      2
      ;


      {\displaystyle 14=7\times 2;}





      49
      =
      7
      ×
      7
      ;


      {\displaystyle 49=7\times 7;}






      21
      =
      7
      ×
      (

      3
      )
      ;


      {\displaystyle -21=7\times (-3);}





      0
      =
      7
      ×
      0
      ;


      {\displaystyle 0=7\times 0;}





      3
      =
      7
      ×
      (
      3

      /

      7
      )
      ,

      3

      /

      7


      {\displaystyle 3=7\times (3/7),\quad 3/7}

      is not an integer;





      6
      =
      7
      ×
      (

      6

      /

      7
      )
      ,


      6

      /

      7


      {\displaystyle -6=7\times (-6/7),\quad -6/7}

      is not an integer.


      Properties


      0 is a multiple of every number (



      0
      =
      0

      b


      {\displaystyle 0=0\cdot b}

      ).
      The product of any integer



      n


      {\displaystyle n}

      and any integer is a multiple of



      n


      {\displaystyle n}

      . In particular,



      n


      {\displaystyle n}

      , which is equal to



      n
      ×
      1


      {\displaystyle n\times 1}

      , is a multiple of



      n


      {\displaystyle n}

      (every integer is a multiple of itself), since 1 is an integer.
      If



      a


      {\displaystyle a}

      and



      b


      {\displaystyle b}

      are multiples of



      x
      ,


      {\displaystyle x,}

      then



      a
      +
      b


      {\displaystyle a+b}

      and



      a

      b


      {\displaystyle a-b}

      are also multiples of



      x


      {\displaystyle x}

      .


      Submultiple


      In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a=b/n) or, equivalently, "b being an integer multiple n of a" (b=n a). This terminology is also used with units of measurement (for example by the BIPM and NIST), where a unit submultiple is obtained 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. As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.


      See also


      Unit fraction
      Ideal (ring theory)
      Decimal and SI prefix
      Multiplier (linguistics)


      References

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