1. **State the problem:** We are given the polynomial $$34z^{14} + bz^7 + 70$$ where $b$ is a positive integer. We want to find the greatest possible value of $b$ such that the polynomial has a factor of the form $$qz^7 + r$$ where $q$ and $r$ are positive integers. 2. **Understand the factorization:** If $$qz^7 + r$$ is a factor, then the polynomial can be written as: $$\left(qz^7 + r\right)\left(sz^7 + t\right)$$ where $s$ and $t$ are positive integers (since the polynomial is degree 14, the other factor must also be degree 7). 3. **Expand the product:** $$\left(qz^7 + r\right)\left(sz^7 + t\right) = qsz^{14} + (qt + rs)z^7 + rt$$ 4. **Match coefficients with the original polynomial:** Comparing with $$34z^{14} + bz^7 + 70$$, we get the system: $$q s = 34$$ $$q t + r s = b$$ $$r t = 70$$ 5. **Find positive integer factor pairs:** - For $q s = 34$, since 34 is $2 \times 17$, possible pairs are $(q,s) = (1,34), (2,17), (17,2), (34,1)$. - For $r t = 70$, since 70 factors as $1 \times 70, 2 \times 35, 5 \times 14, 7 \times 10$ and their reverses. 6. **Calculate $b = q t + r s$ for each combination and find the maximum:** Try each $(q,s)$ pair and each $(r,t)$ pair: - For $(q,s) = (1,34)$: - $(r,t) = (1,70)$: $b = 1 \times 70 + 1 \times 34 = 104$ - $(r,t) = (2,35)$: $b = 1 \times 35 + 2 \times 34 = 35 + 68 = 103$ - $(r,t) = (5,14)$: $b = 1 \times 14 + 5 \times 34 = 14 + 170 = 184$ - $(r,t) = (7,10)$: $b = 1 \times 10 + 7 \times 34 = 10 + 238 = 248$ - For $(q,s) = (2,17)$: - $(r,t) = (1,70)$: $b = 2 \times 70 + 1 \times 17 = 140 + 17 = 157$ - $(r,t) = (2,35)$: $b = 2 \times 35 + 2 \times 17 = 70 + 34 = 104$ - $(r,t) = (5,14)$: $b = 2 \times 14 + 5 \times 17 = 28 + 85 = 113$ - $(r,t) = (7,10)$: $b = 2 \times 10 + 7 \times 17 = 20 + 119 = 139$ - For $(q,s) = (17,2)$: - $(r,t) = (1,70)$: $b = 17 \times 70 + 1 \times 2 = 1190 + 2 = 1192$ - $(r,t) = (2,35)$: $b = 17 \times 35 + 2 \times 2 = 595 + 4 = 599$ - $(r,t) = (5,14)$: $b = 17 \times 14 + 5 \times 2 = 238 + 10 = 248$ - $(r,t) = (7,10)$: $b = 17 \times 10 + 7 \times 2 = 170 + 14 = 184$ - For $(q,s) = (34,1)$: - $(r,t) = (1,70)$: $b = 34 \times 70 + 1 \times 1 = 2380 + 1 = 2381$ - $(r,t) = (2,35)$: $b = 34 \times 35 + 2 \times 1 = 1190 + 2 = 1192$ - $(r,t) = (5,14)$: $b = 34 \times 14 + 5 \times 1 = 476 + 5 = 481$ - $(r,t) = (7,10)$: $b = 34 \times 10 + 7 \times 1 = 340 + 7 = 347$ 7. **Find the greatest $b$:** The maximum value of $b$ from all these is $$2381$$ when $(q,s) = (34,1)$ and $(r,t) = (1,70)$. **Final answer:** $$\boxed{2381}$$