<s>[INST] If $a$ , $b$, $c$ are positive integers satisfying the following $(a^2 +2)(b^2+3)(c^2+4)=2014$ What is the value of $ a+b+c $ If $a$ , $b$, $c$ are positive integers satisfying the following
$(a^2 +2)(b^2+3)(c^2+4)=2014$
What is the value  of $ a+b+c $
I need  details  because  i don't  know  how to solve simlar  problems ?
Thank you for your help. 
 [/INST] $2014 = 2*19*53$  
So each of the $a^2 + 2, b^2+3, c^2 + 4$ are some combination of $2, 19, 53$.  
As none of them can equal 1, the must be that one of them equals $2$, another $19$ and the third $53$.  What possible numbers work?  (Note: just by looking only one can be small enough to equal $2$.)
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