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Question Number 146488 by mathdanisur last updated on 13/Jul/21

$$ifx+y=216and\mathit{d}\mathit{c}\mathit{m}(x;y)=18$$$$findx-y=?$$

Commented by mathdanisur last updated on 13/Jul/21

$$thebiggestcommondivisot$$

Commented by gsk2684 last updated on 13/Jul/21

$$dcmmeans?$$

Answered by mathmax by abdo last updated on 13/Jul/21

$$\mathrm{x}=\mathrm{qd}\mathrm{and}\mathrm{y}={\mathrm{q}}^{{}^{\prime }}\mathrm{d}\mathrm{with}\mathrm{\Delta }(\mathrm{q},{\mathrm{q}}^{{}^{\prime }})=1(\mathrm{d}=18)$$$$\mathrm{x}+\mathrm{y}=216\Rightarrow \mathrm{qd}+{\mathrm{q}}^{{}^{\prime }}\mathrm{d}=216\Rightarrow \mathrm{q}+{\mathrm{q}}^{{}^{\prime }}=\frac{216}{18}=13$$$$\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{system}\{\begin{array}{l}\mathrm{q}+{\mathrm{q}}^{{}^{\prime }}=13\\ \mathrm{\Delta }(\mathrm{q},{\mathrm{q}}^{{}^{\prime }})=1\end{array}$$$$\mathrm{q}=1\Rightarrow {\mathrm{q}}^{{}^{\prime }}=12{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=18\mathrm{and}\mathrm{y}=12\times 18=...$$$$\mathrm{q}=2\Rightarrow {\mathrm{q}}^{{}^{\prime }}=11{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=36\mathrm{and}\mathrm{y}=11\times 18=...$$$$\mathrm{q}=3\Rightarrow {\mathrm{q}}^{{}^{\prime }}=8{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=3\times 18\mathrm{and}\mathrm{y}=8\times 18$$$$\mathrm{q}=4\Rightarrow {\mathrm{q}}^{{}^{\prime }}=9{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=4\times 18\mathrm{and}\mathrm{y}=9\times 18$$$$\mathrm{q}=5\Rightarrow {\mathrm{q}}^{{}^{\prime }}=8{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=5\times 18\mathrm{and}\mathrm{y}=8\times 18$$$$\mathrm{q}=6\Rightarrow {\mathrm{q}}^{{}^{\prime }}=7{\mathrm{sol}}^{\mathrm{o}}\Rightarrow \mathrm{x}=6\times 18\mathrm{and}\mathrm{y}=7\times 18$$$$\mathrm{now}\mathrm{you}\mathrm{can}\mathrm{find}\mathrm{x}-\mathrm{y}....(\mathrm{dont}\mathrm{forget}\mathrm{symetrie}\mathrm{of}\mathrm{the}\mathrm{system}!)$$$$$$$$$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer,butanswer90;126$$

Commented by Rasheed.Sindhi last updated on 15/Jul/21

$$mathmaxsir$$$$In2ndlineatypo:$$$$\mathrm{x}+\mathrm{y}=216\Rightarrow \mathrm{qd}+{\mathrm{q}}^{{}^{\prime }}\mathrm{d}=216\Rightarrow \mathrm{q}+{\mathrm{q}}^{{}^{\prime }}=\frac{216}{18}=12$$

Answered by Rasheed.Sindhi last updated on 13/Jul/21

$$x+y=216and\mathit{g}\mathit{c}\mathit{d}(x,y)=18$$$$Letx=18u\mathrm{\&}y=18vwhere\mathit{g}\mathit{c}\mathit{d}(u,v)=1$$$$x+y=216\Rightarrow 18u+18v=216$$$$\Rightarrow u+v=12$$$$(u,v)=(1,11)=(5,7)=(7,5)=(11,1)$$$$u-v=-10,-2,2,10$$$$x-y=18u-18v=18(u-v)$$$$=18(-10),18(-2),18\left(2\right),18\left(10\right)$$$$x-y=-180,-36,36,180$$

Commented by mathdanisur last updated on 13/Jul/21

$$thanksSer,but90;126$$

Commented by Rasheed.Sindhi last updated on 13/Jul/21

$$Youranswersseemwrongtome.$$$$x+y=216\left(given\right)$$$$Ifx-y=90\left(youranswer\right)$$$$x=153,y=63$$$$\mathit{g}\mathit{c}\mathit{d}(153,63)=9\ne 18$$$$Similarly$$$$x+y=216\left(given\right)$$$$Ifx-y=126\left(youranswer\right)$$$$x=171,y=45$$$$\mathit{g}\mathit{c}\mathit{d}(171,45)=9\ne 18$$

Commented by mathdanisur last updated on 13/Jul/21

$$goodSer,thanks$$

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