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LCM






Mathematica Notation

Traditional Notation









Integer Functions > LCM[n1,n2,...,nm] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/04.10.27.0003.01









  


  










Input Form





GCD[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] == Product[Subscript[n, Subscript[k, 1]] Product[LCM[Subscript[n, Subscript[k, 1]], Subscript[n, Subscript[k, 2]], Subscript[n, Subscript[k, 3]]] \[Ellipsis], {Subscript[k, 1], 1, m}, {Subscript[k, 2], Subscript[k, 1] + 1, m}, {Subscript[k, 3], Subscript[k, 2] + 1, m}], {Subscript[k, 1], 1, m}]/ Product[LCM[Subscript[n, Subscript[k, 1]], Subscript[n, Subscript[k, 2]]] Product[LCM[Subscript[n, Subscript[k, 1]], Subscript[n, Subscript[k, 2]], Subscript[n, Subscript[k, 3]], Subscript[n, Subscript[k, 4]]] \[Ellipsis], {Subscript[k, 1], 1, m}, {Subscript[k, 2], Subscript[k, 1] + 1, m}, {Subscript[k, 3], Subscript[k, 2] + 1, m}, {Subscript[k, 4], Subscript[k, 3] + 1, m}], {Subscript[k, 1], 1, m}, {Subscript[k, 2], Subscript[k, 1] + 1, m}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[SubscriptBox["n", SubscriptBox["k", "1"]], "\[Times]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "3"], "=", RowBox[List[SubscriptBox["k", "2"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", SubscriptBox["k", "1"]], ",", SubscriptBox["n", SubscriptBox["k", "2"]], ",", SubscriptBox["n", SubscriptBox["k", "3"]]]], "]"]], "\[Times]", "\[Ellipsis]"]]]]]]]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", SubscriptBox["k", "1"]], ",", SubscriptBox["n", SubscriptBox["k", "2"]]]], "]"]], "\[Times]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "3"], "=", RowBox[List[SubscriptBox["k", "2"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "4"], "=", RowBox[List[SubscriptBox["k", "3"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", SubscriptBox["k", "1"]], ",", SubscriptBox["n", SubscriptBox["k", "2"]], ",", SubscriptBox["n", SubscriptBox["k", "3"]], ",", SubscriptBox["n", SubscriptBox["k", "4"]]]], "]"]], "\[Times]", "\[Ellipsis]"]]]]]]]]]]]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> gcd </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> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mo> &#8230; </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> k </mi> <mn> 4 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> k </mi> <mn> 4 </mn> </msub> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mo> &#8230; </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <gcd /> <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> <apply> <times /> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 4 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[SubscriptBox["nn", SubscriptBox["k", "1"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "3"], "=", RowBox[List[SubscriptBox["k", "2"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["nn", SubscriptBox["k", "1"]], ",", SubscriptBox["nn", SubscriptBox["k", "2"]], ",", SubscriptBox["nn", SubscriptBox["k", "3"]]]], "]"]], " ", "\[Ellipsis]"]]]]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["nn", SubscriptBox["k", "1"]], ",", SubscriptBox["nn", SubscriptBox["k", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List[SubscriptBox["k", "1"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "3"], "=", RowBox[List[SubscriptBox["k", "2"], "+", "1"]]]], "m"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["k", "4"], "=", RowBox[List[SubscriptBox["k", "3"], "+", "1"]]]], "m"], RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["nn", SubscriptBox["k", "1"]], ",", SubscriptBox["nn", SubscriptBox["k", "2"]], ",", SubscriptBox["nn", SubscriptBox["k", "3"]], ",", SubscriptBox["nn", SubscriptBox["k", "4"]]]], "]"]], " ", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]










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





2001-10-29





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