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Find the sum of the $x$-coordinates of all possible positive integer solutions to $\frac1x+\frac1y=\frac17$.

 May 11, 2019
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For the equation:  1/x  +  1/y  =  1/7,

I believe that there are  3  positive integer solutions:  (8, 56)   (14, 14)   and  (56, 8).

Adding  8 + 14 + 56  =  78.

 

My analysis:  1/x  +  1/y  =  1/7     --->    multiplying by  7xy     --->     7x + 7y  =  xy

Solving for y:     7y - xy  =  -7x     --->     y(7 - x)  =  -7x     --->     y  =  (-7x) / (7 - x)     --->     y  =  (7x) / (x - 7)

Using this equation, as x gets larger and larger,  y  approaches  7, but is always greater than  7.

 

Similarly, solving for x:     x  =  (7y) / (y - 7)

So, as  y  gets largr and larger, x  approaches  7,  but is always greater than  7.

 

So,  8  is the smallest possible value for either  x  or  y.

When  x  =  8,  y  =  56.

This makes  56  the largest possible value.

Trying values between  8  and  56  gives the other solution,  14.

(This solution can be determined by a different analysis.)

 May 11, 2019

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