Question

Aflați cifrele x și y știind că x supra y egal y,x​

Answer 1

$\displaystyle\\\frac{x}{y}=\overline{y,x} \Leftrightarrow \frac{x}{y}=\frac{\overline{yx}}{10}\Leftrightarrow 10x=y(10y+x),~cum~10\leqslant 10x\leqslant 90 \implies\\10\leqslant y(10y+x)\leqslant 90 \Leftrightarrow 10\leqslant 10y^2+xy\leqslant 90,~de~unde~10\leqslant 10y^2 \leqslant 90~|:10,\\1\leqslant y^2\leqslant 10\implies y\in\left\{1,2,3\right\}.$

$\displaystyle\\Mai~mult,~10y^2+xy~este~\mathcal{M}_{10},~dar~10y^2~este~\mathcal{M}_{10}\implies~xy~este~\mathcal{M}_{10},~\\dar~cum~y\in\left\{1,2,3\right\}~iar~xy~din~cauza~asta~va~fi~cel~mult~27\implies \\xy~este~10~sau~20,~dar~evident~20~nu~poate~fi,~deci~xy=10~de~unde~\\\boxed{y=2}~iar~\boxed{x=5},~pentru~ca~10=2\cdot5,~iar~y~nu~poate~fi~mai~mare~decat~3.$