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GCD






Mathematica Notation

Traditional Notation









Integer Functions > GCD[n1,n2,...,nm] > Inequalities





http://functions.wolfram.com/04.08.29.0001.01









  


  










Input Form





GCD[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, k]] LCM[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, k]]^ (k - 1) <= Product[Subscript[n, j], {j, 1, k}] <= GCD[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, k]]^(k - 1) LCM[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, k]] /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] > 0 && Element[Subscript[n, 2], Integers] && Subscript[n, 2] > 0 && \[Ellipsis] && Element[Subscript[n, k], \[DoubleStruckCapitalZ]] && Subscript[n, k] > 0 && Element[k, Integers] && k > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], " ", SuperscriptBox[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], RowBox[List["k", "-", "1"]]]]], "\[LessEqual]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "k"], SubscriptBox["n", "j"]]], "\[LessEqual]", RowBox[List[SuperscriptBox[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "1"], ">", "0"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], ">", "0"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], ">", "0"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]










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> <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> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <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> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> &#8804; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> &#8804; </mo> <mrow> <msup> <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> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <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> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <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> k </ci> </apply> </apply> <apply> <power /> <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> k </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <power /> <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> k </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <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> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <ci> &#8230; </ci> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], " ", SuperscriptBox[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], RowBox[List["k", "-", "1"]]]]], "\[LessEqual]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "k"], SubscriptBox["n", "j"]]], "\[LessEqual]", RowBox[List[SuperscriptBox[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "k"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "1"], ">", "0"]], "&&", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "2"], ">", "0"]], "&&", "\[Ellipsis]", "&&", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "\[DoubleStruckCapitalZ]"]], "&&", RowBox[List[SubscriptBox["n", "k"], ">", "0"]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]










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





2002-12-18





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