Lời giải:
Nếu $2x+3y\vdots 17$
$\Rightarrow 9(2x+3y)\vdots 17$
$\Rightarrow 18x+27y\vdots 17$
$\Rightarrow 18x+27y-17y\vdots 17$
$\Rightarrow 18x+10y\vdots 17$
$\Rightarrow 2(9x+5y)\vdots 17$
$\Rightarrow 9x+5y\vdots 17(1)$
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Nếu $9x+5y\vdots 17$
$\Rightarrow 2(9x+5y)\vdots 17$
$\Rightarrow 18x+10y\vdots 17$
$\Rightarrow 18x+10y+17y\vdots 17$
$\Rightarrow 18x+27y\vdots 17$
$\Rightarrow 9(2x+3y)\vdots 17$
$\Rightarrow 2x+3y\vdots 17(2)$
Từ $(1); (2)$ ta có đpcm.