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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
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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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