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Given positive integers x and $y$ such that $x\neq y$ and 1/x + 1/y = 1/8, what is the smallest possible value for x + y?

 Feb 15, 2022
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1/x + 1/y  =  1/8     --->     1/y  =  1/8 - 1/x

                                         1/y  =  (x - 8) / 8x

                                            y  =  8x / (x - 8)

 

x + y  =  x + 8x / (x - 8)     --->     x + y  =  [ x(x - 8) ] / (x - 8) + 8x / (x - 8)

                                                              =  (x2 - 8x + 8x) / (x - 8)

                                                              =  x2 / (x - 8)

 

For values between 0 and 8, the value of x + y is negative.

For values above 8, the value is positive and increasing.

 

Thus, the smallest value of x + y occurs at the integer value x = 9.

 

You can find this value ...

 Feb 15, 2022

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