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http://functions.wolfram.com/04.08.16.0004.01
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GCD[\[Mu] m, \[Nu] n] == GCD[m, n] GCD[\[Mu], \[Nu]]
GCD[m/GCD[m, n], \[Nu]/GCD[\[Mu], \[Nu]]] GCD[n/GCD[m, n],
\[Mu]/GCD[\[Mu], \[Nu]]] /; Element[m, Integers] && m > 0 &&
Element[n, Integers] && n > 0 && Element[\[Mu], Integers] && \[Mu] > 0 &&
Element[\[Nu], Integers] && \[Nu] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List[RowBox[List["\[Mu]", " ", "m"]], ",", RowBox[List["\[Nu]", " ", "n"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]], RowBox[List["GCD", "[", RowBox[List[FractionBox["m", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]], ",", FractionBox["\[Nu]", RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]]]]], "]"]], RowBox[List["GCD", "[", RowBox[List[FractionBox["n", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]], ",", FractionBox["\[Mu]", RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Mu]", ">", "0"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> , </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> m </mi> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> , </mo> <mfrac> <mi> ν </mi> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> , </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> , </mo> <mfrac> <mi> μ </mi> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> , </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> μ </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <gcd /> <apply> <times /> <ci> μ </ci> <ci> m </ci> </apply> <apply> <times /> <ci> ν </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <apply> <gcd /> <ci> μ </ci> <ci> ν </ci> </apply> <apply> <gcd /> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <apply> <gcd /> <ci> μ </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gcd /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <apply> <gcd /> <ci> μ </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> μ </ci> <integers /> </apply> <apply> <in /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GCD", "[", RowBox[List[RowBox[List["\[Mu]_", " ", "m_"]], ",", RowBox[List["\[Nu]_", " ", "n_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], " ", RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]], " ", RowBox[List["GCD", "[", RowBox[List[FractionBox["m", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]], ",", FractionBox["\[Nu]", RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]]]]], "]"]], " ", RowBox[List["GCD", "[", RowBox[List[FractionBox["n", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]], ",", FractionBox["\[Mu]", RowBox[List["GCD", "[", RowBox[List["\[Mu]", ",", "\[Nu]"]], "]"]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Mu]", ">", "0"]], "&&", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
