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How many pairs of positive integers \((a,b)\) satisfy \(\frac{1}{a} + \frac{1}{b}=\frac{2}{17}\)?

 Apr 9, 2020
 #1
avatar+4569 
+3

1/a + 1/b = 2/17

 

b/ab+a/ab=2/17

 

a+b/ab=2/17

17(a+b)=2ab

 

17a+17b=2ab

 

 

2ab-17a-17b=0

 

Multiply the equation by two...

 

4ab-34a-34b=0

 

If 289(17^2) is added to both sides, then (2a-17)(2b-17)=289.

 

289=1*289 and 17*17.

 

2a-17=1  

 

2a=18   a=9              2b-17=289   2b=153             (9,153), (153,9)

 

 

2a-17=17  2a=34  a=17            2b-17=17,  2b=34, b=17            (17,17)

Thus, there should be three(3) positive ordered pairs.

 Apr 9, 2020
 #2
avatar+337 
+2

Thank you!

 Apr 10, 2020

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