Integrate[(FractionalPart[t/a] - 1/2) (FractionalPart[t/b] - 1/2), {t, 0, a b}] == (a b GCD[a, b])/(12 LCM[a, b]) /; Element[a, Integers] && Element[b, Integers]

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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> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> t </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> t </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <ci> a </ci> <ci> b </ci> </apply> </uplimit> <apply> <times /> <apply> <plus /> <apply> <ci> FractionalPart </ci> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> FractionalPart </ci> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <gcd /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <lcm /> <ci> a </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> a </ci> <integers /> </apply> <apply> <in /> <ci> b </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29