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# PLS HELP CPHILL

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Given positive integers  and  such that  and , what is the smallest possible value for ?$$Given positive integers x and y such that x\neq y and \frac{1}{x} + \frac{1}{y} = \frac{1}{15}, what is the smallest possible value for x + y?$$

Jan 29, 2024

#1
+129742
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1/x  + 1/y =  1/15

( x + y)  / (xy)  = 1/15

(xy)    / ( x + y)  =  15

xy =  15 ( x + y)

xy  =15x + 15y

xy - 15y =  15x

y ( x - 15)  =  15x

y =   15x / ( x - 15)

x          y

16      240

18       90

20       60

24       40

30       30

40       24

Smallest  x + y =  40 + 24  =  64

Jan 29, 2024
#2
+944
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30 + 30 is 60

.

Bosco  Jan 29, 2024
#3
+129742
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True....but i'm assuming that x and y are different integers.....

CPhill  Jan 30, 2024
#4
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Chris, you were right all along.  The problem stipulates x not equal to y.

I had overlooked that in all that run-together text and those dollar signs.

After I realized it, too late to edit, I was relieved that my comment got lost

in the moderation by AI, which btw I maintain is a little short on the "I" part.

Bosco  Feb 1, 2024