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Given positive integers x and y such that x is not equal to y and 1/x + 1/y = 1/24, what is the smallest possible positive value for x + y?

 Nov 26, 2020
 #1
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x = 42  and  y = 56

x + y = 42 + 56 =98 - smallest positive value.

 Nov 27, 2020
 #2
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Let z =  24

Then   x  must  be   z + a   where  a > 0

And y  must be   z + b   where b > 0     ....so we have

 

1/ (z + a)    +  1 / (z + b)   =     1/ z      

 

( z + a + z + b)  / [ (z + a) (z + b) ]  =  1/ z

 

( a + b + 2z)  / ( z^2 + az + bz + ab)  = 1 / z      cross-multiply

 

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

 

z^2  =  ab

 

24^2  =  ab

 

576  =  ab

 

Factors of 576  =

 

1 | 2 | 3 | 4 | 6 | 8 | 9 | 12 | 16 | 18 | 24 | 32 | 36 | 48 | 64 | 72 | 96 | 144 | 192 | 288 | 576 

 

18 and 32    are  the two factors  that will lead to minimum values  for  x and  y

 

Let a = 18   and  b = 32

 

x =  z + a =   24 + 18  =  42

y =  z + b =   24  + 32  =  56

 

x + y =  98

 

cool cool cool

 Nov 27, 2020
edited by CPhill  Nov 27, 2020

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