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Find all integers x and y such that x + y = xy.

 May 28, 2020
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Find all integers x and y such that \(x + y = xy\).

 

\(\begin{array}{|rcll|} \hline \mathbf{ x + y} &=& \mathbf{ = xy} \\ xy-x-y &=& 0 \\ x(y-1)-y &=& 0 \quad | \quad +1 \\ x(y-1)-y+1 &=& 1 \\ x(y-1)-(y-1) &=& 1 \\ (x-1)(y-1) &=& 1 \\ \mathbf{(x-1)(y-1)} &=& \mathbf{1} \\ \hline \end{array}\)

 

Solutions:

\(\begin{array}{|rcll|} \hline 1) & 1 &=& 1*1 \\ 2) & 1 &=& (-1)(-1) \\ \hline \end{array}\)

 

\(\begin{array}{|r|r|r|l|} \hline (x-1) & (y-1) & x & y & \text{$x$ is integer and $y$ is integer}\\ \hline 1 & 1 & x-1=1 & y-1 = 1 \\ & & x =2 & y =2 & \checkmark \\ \hline -1 & -1 & x-1=-1 & y-1 = -1 \\ & & x =0 & y =0 \\ \hline \end{array} \)

 

laugh

 May 28, 2020

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