Multiple (mathematics)

<h2 id="examples">Examples</h2>
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.
<h2 id="properties">Properties</h2>
<ul><li>0 is a multiple of every number (
 
 
 
 0
 =
 0
 ⋅
 b
 
 
 {\displaystyle 0=0\cdot b}
 
).</li>
<li>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.</li>
<li>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}
 
.</li></ul>
<h2 id="submultiple">Submultiple</h2>
In some texts[which?], "a is a submultiple of b" has the meaning of "a being a <a href="/facts/Unit_fraction/tjAfHzCH">unit fraction</a> of b" (a=b/n) or, equivalently, "b being an <a href="/facts/Integer_multiple/tkte6NXa">integer multiple</a> n of a" (b=n a). This terminology is also used with <a href="/facts/Units_of_measurement/pvTpWbmC">units of measurement</a> (for example by the <a href="/facts/BIPM/tPEu4X0s">BIPM</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> and <a href="/facts/NIST/gr9Fnh85">NIST</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>), where a <a href="/facts/Unit_submultiple/7d7Uqd9N">unit submultiple</a> is obtained by <a href="/facts/Prefix/C4Lw8npg">prefixing</a> the main unit, defined as the quotient of the main unit by an integer, mostly a power of 103. For example, a <a href="/facts/Millimetre/25mdTXpa">millimetre</a> is the 1000-fold submultiple of a <a href="/facts/Metre/hPhzc1CV">metre</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> As another example, one <a href="/facts/Inch/CocFfm8p">inch</a> may be considered as a 12-fold submultiple of a <a href="/facts/Foot_(unit)/XKamEGjV">foot</a>, or a 36-fold submultiple of a <a href="/facts/Yard_(unit)/ddkwJz1q">yard</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Unit_fraction/tjAfHzCH">Unit fraction</a></li>
<li><a href="/facts/Ideal_(ring_theory)/vCvytduX">Ideal (ring theory)</a></li>
<li><a href="/facts/Decimal/7omfVNIM">Decimal</a> and <a href="/facts/SI_prefix/Y2MzkaeL">SI prefix</a></li>
<li><a href="/facts/Multiplier_(linguistics)/YTu1BESm">Multiplier (linguistics)</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric W. "Multiple". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16. <a href="92-822-2213-6" target="_blank">92-822-2213-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes. <a href="http://physics.nist.gov/Pubs/SP811/sec04.html" target="_blank">http://physics.nist.gov/Pubs/SP811/sec04.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16. <a href="92-822-2213-6" target="_blank">92-822-2213-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes. <a href="http://physics.nist.gov/Pubs/SP811/sec04.html" target="_blank">http://physics.nist.gov/Pubs/SP811/sec04.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Multiple (mathematics) open-in-new

Multiple (mathematics)