Exercice AA-002-NT


  • Math&Maroc

    aa et bb sont des entiers tels que (a2+b2)(2a3+b2)(a^2+b^2)\vert (2a^3+b^2).

    Montrer que (a2+b2)(2a3b2+ab2+3b4)(a^2+b^2)\vert (2a^3b^2+ab^2+3b^4).



  • @Mohammed-Aassila
    on a a2+b2a^{2}+b^{2} divise (a2+b2)(2a+1)(a^{2}+b^{2})(2a+1) donc a2+b2a^{2}+b^{2} divise a2+2ab2a^{2}+2ab^{2}

    d ou a2+b2a^{2}+b^{2} divise a3+2a2b2a^{3}+2a^{2}b^{2} et on a a2+b2a^{2}+b^{2} divise (a2+b2)(a+2b2)(a^{2}+b^{2})(a+2b^{2})
    ce qui donne a2+b2a^{2}+b^{2} divise ab2+2b4ab^{2}+2b^{4} et puisque a2+b2a^{2}+b^{2} divise 2a3b2+b42a^{3}b^{2}+b^{4}
    alors a2+b2a^{2}+b^{2} divise 2a3b2+ab2+3b42a^{3}b^{2}+ab^{2}+3b^{4}



  • On a a2+b2a^2 + b^2 \vert (2a32a^3+b2b^2).aa donc a2+b2a^2 + b^2 \vert 2a4+ab2a^4 +ab^2
    et on a a2+b2a^2 + b^2 \vert (2a3+b22a^3+b^2).b2b^2 donc a2+b2a^2 + b^2 \vert 2a3b2+b4a^3b^2 + b^4
    donc a2+b2a^2 + b^2 \vert 2a3b2+b4+ab2+2a42a^3b^2 + b^4 + ab^2 + 2a^4
    d'où a2+b2a^2 + b^2 \vert 2a3b2+ab2+3b42(b4a4)2a^3b^2 + ab^2 + 3b^4 -2(b^4-a^4)
    et b4a4=(b2a2)(a2+b2) b^4 - a^4 = (b^2-a^2)(a^2+b^2)


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