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Given positive integers x and y such that x is unequal to y and 1/x+1/y=1/18, what is the smallest possible value for x+y?

 Jan 15, 2020
edited by Guest  Jan 15, 2020
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1/x + 1/y  =  1/18

 

Let  z  =  18

 

And   x, y  must be >  18

 

So  

 

Let x  =  z + a         and Let y  =  z + b

 

So we have that

 

  1              1              1

____  +   ____  =   _____     simplify

z + a       z + b           z

 

[ z + b + z + a]            1

____________ =   _______

[ z + a] [z + b]              z

 

 

[2z + a + b]                   1

_____________  =   _____               cross-multiply

 [ z+ a ] [ z + b]            z 

 

z [ 2z + a + b ]   =  [ z + a] [ z + b ]

 

2z^2 +az + bz  =    [ z + a ] [ z + b ]

 

2z^2 + az + bz   =  z^2  + az + bz + ab

 

z^2  =  ab

 

(18)^2  =  ab

 

324  =  ab

 

The factors  of  324   are  

 

1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 27 | 36 | 54 | 81 | 108 | 162 | 324 

 

The smallest possible value for a,b    is   12 , 27

 

So

 

x =  z + a  =   18 + 12  = 30

y =  z + b  =   18 + 27  = 45

 

So....the minimum value for x + y  =  75

 

 

cool cool cool

 Jan 15, 2020

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