tim 2 số a,b nguyên dương sao cho $\dfrac{a^{2}-2}{ab+2}$ nguyên
$\dfrac{a^{2}-2}{ab+2}$
Started By taubietrui, 18-12-2011 - 18:13
#2
Posted 18-12-2011 - 22:28
$$(x^2-2) \vdots (xy+1)$$
$$ \implies y(x^2-2) \vdots (xy+2)$$
$$ \implies \left( x(xy+2)-2(x+y) \right) \vdots (xy+2)$$
$$ \implies 2(x+y) \vdots (xy+2)$$
Vì $x,y$ nguyên dương nên
$$xy+2 \le 2x+2y \implies y(x-2) \le 2x-2$$
Ta xét $x=1,2$ và $x \ge 3$.
Từ đây tìm ra $x=4,y=3$.
$$ \implies y(x^2-2) \vdots (xy+2)$$
$$ \implies \left( x(xy+2)-2(x+y) \right) \vdots (xy+2)$$
$$ \implies 2(x+y) \vdots (xy+2)$$
Vì $x,y$ nguyên dương nên
$$xy+2 \le 2x+2y \implies y(x-2) \le 2x-2$$
Ta xét $x=1,2$ và $x \ge 3$.
Từ đây tìm ra $x=4,y=3$.
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#3
Posted 18-10-2017 - 19:46
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