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Question Number 102588 by bobhans last updated on 10/Jul/20

$$solve109x+103y=5forx,yareinteger$$

Commented by mr W last updated on 10/Jul/20

$$seeQ44819$$

Commented by mr W last updated on 10/Jul/20

$$seealsoQ19198$$

Commented by bobhans last updated on 10/Jul/20

$$collsir$$

Answered by 1549442205 last updated on 10/Jul/20

$$\mathrm{y}=\frac{5-109\mathrm{x}}{103}=\frac{5-6\mathrm{x}}{103}-\mathrm{x}.\mathrm{Put}\frac{5-6\mathrm{x}}{103}=\mathrm{a}(\mathrm{a}\in \mathbb{Z})$$$$\Rightarrow 5-6\mathrm{x}=103\mathrm{a}\iff \mathrm{x}=\frac{5-103\mathrm{a}}{6}=-17\mathrm{a}+\frac{5-\mathrm{a}}{6}$$$$\mathrm{Put}\frac{5-\mathrm{a}}{6}=\mathrm{b}\Rightarrow \mathrm{a}=5-6\mathrm{b}.\mathrm{From}\mathrm{this}\mathrm{we}$$$$\mathrm{get}\mathrm{x}=-17\mathrm{a}+\mathrm{b}=103\mathrm{b}-85,\mathrm{y}=-\mathrm{x}+\mathrm{a}$$$$=-109\mathrm{b}+90.\mathrm{Thus},\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{eqs}\mathrm{be}$$$$\{\begin{array}{l}\mathbf{x}=103\mathbf{b}-85\\ \mathbf{y}=-109\mathbf{b}+90\end{array}(\mathbf{b}\in \mathbb{Z})$$$$$$

Answered by bemath last updated on 10/Jul/20

$$\{\begin{array}{l}109=1\left(103\right)+6\\ 103=17\left(6\right)+1\end{array}$$$$\{\begin{array}{l}103-17\left(6\right)=1\\ 103-17(109-103)=1\end{array}$$$$\iff 109(-17)+103\left(18\right)=1...\left(\times 5\right)$$$$109(-85)+103\left(90\right)=5$$$$wegetgenerallsolution$$$$x=-85+103n$$$$y=90-109n,n\in \mathbb{Z}$$

Answered by PRITHWISH SEN 2 last updated on 10/Jul/20

$$109\mathrm{x}+103\mathrm{y}=5\mathbf{It}\mathbf{is}\mathbf{a}\mathbf{diophantine}\mathbf{equation}$$$$\because \gcd (109,103)=1$$$$\mathrm{then}\mathrm{there}\mathrm{exists}\mathrm{two}\mathrm{integers}\mathrm{u}\mathrm{and}\mathrm{v}\mathrm{such}\mathrm{that}$$$$109\mathrm{u}+103\mathrm{v}=1$$$$\mathrm{here}\mathrm{u}=-85\mathrm{amd}\mathrm{v}=90$$$$\mathrm{then}109\mathrm{x}+103\mathrm{y}=-109.85+103.90$$$$\Rightarrow 109(\mathrm{x}+85)=-103(\mathrm{y}-90)$$$$\frac{\mathrm{x}+85}{-103}=\frac{\mathrm{y}-90}{109}=\mathrm{t}\left(\mathbf{integer}\right)$$$$\because \gcd (109,103)=1\mathrm{then}103\mid (\mathrm{x}+85)\mathrm{and}109\mid (\mathrm{y}-90)$$$$\therefore \mathbf{x}=-103\mathbf{t}-85$$$$\mathbf{y}=109\mathbf{t}+90$$$$$$

Answered by bobhans last updated on 10/Jul/20

$$wellhas2solution??$$

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