Question

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Answer 1

 

$\displaystyle\bf\\k\in~Z;~x\in~Z;~y\in~Z\\\\x=\frac{3}{k}+\frac{2}{k-1}\\\\y=\frac{3}{k-2}+\frac{5}{k+2}\\\\\text{Pentru ca x si y sa fie intregi, trebuie ca fecare fractie sa fie numar intreg.}\\\text{Rezulta ca fiecare numitor trebuie sa fie divizor al numaratorului.}\\\\Fractia~1:\\k\in\{-3,-1,1,3\}\\\\Fractia~2:\\(k-1)\in\{-2,-1,1,2\}\implies~k\in\{-1,0,2,3\}\\\\Fractia~3:\\(k-2)\in\{-3,-1,1,3\}\implies~k\in\{-1,1,3,5\}\\\\Fractia~4:\\(k+2)\in\{-5,-1,1,5\}\implies~k\in\{-7,-3,-1,3\}$

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$\displaystyle\bf\\\text{Intersectam multimile.}\\\\\{-3,-1,1,3\}\bigcap\{-1,0,2,3\}\bigcap\{-1,1,3,5\}\bigcap\{-7,-3,-1,3\}=\{-1,3\}\\\\k\in\{-1,3\}\\\\\implies~x~si~y~\in~Z~daca~k=-1~sau~k=3\\\\Verificare:\\\\k=-1\\\\x=\frac{3}{-1}+\frac{2}{-1-1}=-3+(-1)=-4\in~N\\\\y=\frac{3}{-1-2}+\frac{5}{-1+2}=-1+5=4\in~Z\\\\k=3\\\\x=\frac{3}{3}+\frac{2}{3-1}=1+1=2\in~Z\\\\y=\frac{3}{3-2}+\frac{5}{3+2}=3+1=4\in~Z$