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In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{30}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $k$?

 Aug 4, 2023
 #2
avatar+126978 
+1

This problem is WAY more involved than you may think!!

 

1/j  + 1/k   =  1/30

 

Let  30  =  z

 

j and  k  must  be >  30

 

So

 

Let   j  =  z + m      and   k =   z  +  n

 

So  we  have

 

1/ ( z + m)  +  1/ ( z + n)  =  1/z

 

(z +n  + z + m)            1

____________  =     ____                 cross-multiply

(z + m) (z + n)             z

 

 

(m + n + 2z) z  =    (z + m) (z + n)

 

2z^2  + zm + zn  =  z^2  + zm + zn  +  mn

 

z^2    =   mn

 

30^2 =  mn

 

900  = mn

 

Now....I WILL NOT list all the  integer possibilities for this problem but here's a start

 

mn = 900

 

So  we have that

 

m    n          j = z + m       k = z + n

1    900             31              930

2   450              32              480

3   300              33              330

4   225              34              255

.

.

.

And so on !!!!!

 

WolframAlpha has done the heavy lifting  here :

 

https://www.wolframalpha.com/input?i=1%2Fj+%2B+1%2Fk+%3D+1%2F30

 

P.S. - I wouldn't even attempt to sum all the positive k"s

 

 

cool cool cool

 Aug 5, 2023
edited by CPhill  Aug 5, 2023

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