Let $r,s$ and $k$ be positive integers such that $k+s+r+rs+sk+rk+rsk=30148$

then find the value of $r+s+k$ .

The answer is 136.

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Adding 1 to both sides and factoring yields $(r+1)(s+1)(k+1)=30149$ . Since $30149=7 \cdot 59 \cdot 73$ and $r, s, k>0$ , we know $(r+1), (s+1), (k+1)$ are each nontrivial factors of $30149$ , so WLOG, $r=6, s=58, k=72$ . Therefore, $r+s+k= \boxed{136}$ .