<s>[INST] Find sum of all integer solution of $\frac{2}{a} + \frac{3}{b} + \frac{4}{c} + \frac{5}{d} = \frac{1}{14}$ Given  $\frac{2}{a} +  \frac{3}{b} + \frac{4}{c} + \frac{5}{d}  = \frac{1}{14}$ , which $a,b,c ,d$ are an positive integer  , all solution of $a,b,c,d$  are members of Set  "S"  ,then find sum all of members in set "S"
I don't know how to start to solve this problem but I've got 2 solution that is  $\frac{2}{52} +  \frac{3}{598} + \frac{4}{437} + \frac{5}{266}  = \frac{1}{14}$  and $\frac{2}{224} +  \frac{3}{304} + \frac{4}{437} + \frac{5}{115}  = \frac{1}{14}$ 
question 2 If problem say   that $a,b,c,d$ are an all integer , they will have negative integer to this solution ? please give me hint or theorem relevant. Thank you in advance .  
 [/INST] For Question 2, there are infinitely many integer solutions.  Take $$(a,b,c,d):=(4,-7,5k,-4k)\,,\text{ where }k\in\mathbb{Z}_{\neq 0}\,.$$
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