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For integers \(a\) and \(b\), find integer solution of equation \(a / x - b/y = 1\)
02-21-2010, 06:29 PM
Post: #1
For integers \(a\) and \(b\), find integer solution of equation \(a / x - b/y = 1\)
If \(a=b=0\) there is no solutions
If \(b = 0\) then \((x,y)=(a,k)\) with \(k\in \mathbf{Z}\) are solutions
Assume that \(b \neq 0\)

Let \(\lambda y = x\), then \((a-\lambda b) = x\), thus for \(k=\lambda b\), we need \(k\) be an integer to ensure \(x\) be an integer. Also, \(ky=b\lambda y=bx=ab-bk\), thus \(k | ab\).
Therefore we have \[\left\{\begin{matrix}
x & = & a-k\\
y & = & \frac{ab}{k}-b
\end{matrix}\right. \]
where \(k \notin \left\{0, a\right \}\) and \(k | ab\)
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