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How to find all the distinct positive integer ordered pairs for $a+b\le100$ such that :

$$\frac{a+\frac{1}{b}}{b+\frac{1}{a}}=10$$

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    $\begingroup$ Simplify the LHS. :) $\endgroup$ – darij grinberg Aug 12 '13 at 22:59
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Hint: $$\frac{a+\frac{1}{b}}{b+\frac{1}{a}}=\frac{a+\frac{1}{b}}{\frac{b}{a}(a+\frac{1}{b})}$$

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  • $\begingroup$ or multiplying top and bottom by $ab (\gt 0 )$ to give $\dfrac{a(ab+1)}{b(ab+1)} = 10$ $\endgroup$ – Henry Aug 12 '13 at 23:23
  • $\begingroup$ @Henry Sure. ${}{}{}{}$ $\endgroup$ – Amr Aug 12 '13 at 23:45

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