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If $a$ and $b$ are integers with $a > b$, what is the smallest possible positive value of $\frac{a+b}{a-b} + \frac{a-b}{a+b}$?

AnonymousConfusedGuy  Oct 13, 2017
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\($\frac{a+b}{a-b} + \frac{a-b}{a+b}$\)

 

[ (a + b)(a + b) + (a - b)(a - b) ]  / [ a^2 - b^2 ] =

 

 ( [ a^2 + 2ab + b^2  + a^2 - 2ab +b^2 ] / [ a^2 - b^2]  =

 

[ 2a^2 + 2b^2 ] / [ a^2 - b^2 ]

 

2 [ a^2 + b^2] / [ a^2 - b^2 ]

 

The smallest positive value, 2, is achieved when  b  = 0  and a is any positive integer

 

In every other case,  [ a^2 + b^2] / [ a^2 - b^2 ] will be > 1   and the expression will be > 2

 

 

cool cool cool

CPhill  Oct 13, 2017
edited by CPhill  Oct 13, 2017

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