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http://functions.wolfram.com/04.08.03.0012.01
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GCD[LCM[k, m], LCM[k, n], LCM[m, n]] == LCM[GCD[k, m], GCD[k, n],
GCD[m, n]] /; Element[k, Integers] && k > 0 && Element[m, Integers] &&
m > 0 && Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List[RowBox[List["LCM", "[", RowBox[List["k", ",", "m"]], "]"]], ",", RowBox[List["LCM", "[", RowBox[List["k", ",", "n"]], "]"]], ",", RowBox[List["LCM", "[", RowBox[List["m", ",", "n"]], "]"]]]], "]"]], "\[Equal]", " ", RowBox[List["LCM", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "m"]], "]"]], ",", RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]], ",", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "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> lcm </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> lcm </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> lcm </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> lcm </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </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> m </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> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <gcd /> <apply> <lcm /> <ci> k </ci> <ci> m </ci> </apply> <apply> <lcm /> <ci> k </ci> <ci> n </ci> </apply> <apply> <lcm /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <lcm /> <apply> <gcd /> <ci> k </ci> <ci> m </ci> </apply> <apply> <gcd /> <ci> k </ci> <ci> n </ci> </apply> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> k </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </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["LCM", "[", RowBox[List["k_", ",", "m_"]], "]"]], ",", RowBox[List["LCM", "[", RowBox[List["k_", ",", "n_"]], "]"]], ",", RowBox[List["LCM", "[", RowBox[List["m_", ",", "n_"]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LCM", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "m"]], "]"]], ",", RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]], ",", RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]]]], "]"]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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