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 $

$2014 = 2*19*53$

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 $2$.)

Try factoring $2014$. Then you might see how to find $a$, $b$, and $c$. If you need more help just ask!