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Question Number 119440 by mathocean1 last updated on 24/Oct/20

$$\mathrm{Determine}\mathrm{the}\mathrm{integer}\mathrm{which}\mathrm{can}$$$$\mathrm{be}\mathrm{written}:\mathrm{N}=\overline{\mathrm{xyz}}{}^7=\overline{\mathrm{zyx}}{}^{11}$$

Commented by mathocean1 last updated on 24/Oct/20

$$\mathrm{Please}\mathrm{can}\mathrm{you}\mathrm{detail}\mathrm{sir}$$

Answered by mr W last updated on 25/Oct/20

$$x,y,zaredigitsofanintegerofbase7,$$$$\Rightarrow 0⩽x,y,z⩽6$$$$besides,x⩾1,z⩾1.$$$$$$$${\left(xyz\right)}_7=49x+7y+z$$$${\left(zyx\right)}_{11}=121z+11y+x$$$$\Rightarrow 49x+7y+z=121z+11y+x$$$$\Rightarrow 48x-4y=120z$$$$\Rightarrow 12x-y=30z$$$$ymustbeamultipleof6,sayy=6Y$$$$\Rightarrow 2x-Y=5z$$$$Y=0or1$$$$withY=0:$$$$2x=5z$$$$\Rightarrow x=5,z=2$$$$withY=1:$$$$2x-1=5z$$$$\Rightarrow x=3,z=1$$$$$$$$thesolutionsare:$$$$x=5,Y=0(i.e.y=0),z=2$$$$\Rightarrow N={\left(502\right)}_7={\left(205\right)}_{11}=247$$$$x=3,Y=1(i.e.y=6),z=1$$$$\Rightarrow N={\left(361\right)}_7={\left(163\right)}_{11}=190$$

Commented by mathocean1 last updated on 25/Oct/20

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}.$$

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