Find all positive integers c such that it is possible to write c = a/b + b/a

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To find all positive integers c such that c = a/b + b/a, it is established that c can be expressed as c = (a^2 + b^2) / (ab), meaning that ab must divide a^2 + b^2. The discussion focuses on identifying the common factors of a and b that allow this condition to hold. The user proposes that c = 2 is a valid solution, seeking confirmation on its correctness. Further exploration of the relationship between a and b may reveal additional positive integer solutions for c. The conversation emphasizes the mathematical relationship and the need for integer solutions.
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Find all positive integers c such that it is possible to write c = a/b + b/a with positive integers a and b.
Please help me :smile:
 
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You know c=\frac{a^2+b^2}{ab} is an integer, so ab divides a^2+b^2. What factors must a and b have in common?
 
i got c=2 as the answer, is it right?
 
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