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LCM






Mathematica Notation

Traditional Notation









Integer Functions > LCM[n1,n2,...,nm] > Primary definition





http://functions.wolfram.com/04.10.02.0002.01









  


  










Input Form





LCM[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] == p /; Element[Re[p], Integers] && Element[Im[p], Integers] && Element[Re[p/Subscript[n, k]], Integers] && Element[Im[p/Subscript[n, k]], Integers] && 1 <= k <= m && ( !Exists[q, Abs[q] < Abs[p] && Element[Re[q], Integers] && Element[Im[q], Integers]] && Element[Re[q/Subscript[n, k]], Integers] && Element[Im[q/Subscript[n, k]], Integers] && 1 <= k <= m)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", "p"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "p", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "p", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", FractionBox["p", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox["p", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Not", "[", RowBox[List["Exists", "[", RowBox[List["q", ",", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", RowBox[List["Abs", "[", "p", "]"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "q", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "q", "]"]], "\[Element]", "Integers"]]]]]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List["Re", "[", FractionBox["q", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox["q", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> p </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> p </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> p </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> m </mi> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#172; </mo> <mrow> <msub> <mo> &#8707; </mo> <mi> q </mi> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> RAbs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> p </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> q </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> q </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <and /> <apply> <in /> <apply> <real /> <ci> p </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> p </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <apply> <times /> <ci> p </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <ci> p </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> m </ci> </apply> <apply> <and /> <apply> <not /> <apply> <exists /> <bvar> <ci> q </ci> </bvar> <apply> <and /> <apply> <lt /> <apply> <ci> RAbs </ci> <ci> q </ci> </apply> <apply> <abs /> <ci> p </ci> </apply> </apply> <apply> <in /> <apply> <real /> <ci> q </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> q </ci> </apply> <integers /> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <real /> <apply> <times /> <ci> q </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <ci> q </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["p", "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "p", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "p", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", FractionBox["p", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox["p", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["!", RowBox[List[SubscriptBox["\[Exists]", "q"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", RowBox[List["Abs", "[", "p", "]"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "q", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "q", "]"]], "\[Element]", "Integers"]]]], ")"]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", FractionBox["q", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox["q", SubscriptBox["n", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]], ")"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





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