There are three rectangular tanks in a building. The length, width and height of the first tank are m meters each, and the length, width and height of the second tank are n meters each. However, the length, width and height of the third tank are m meters, n meters and 1 meter, respectively.Initially, the first tank is full of water, while the second and the third are empty. When the second and the third tanks are completely filled with water transferred from the first tank, 85,000 liters of water is still left in the first tank.
If both m and n are positive integers, what is the value of m? (1 meter$$^{3}$$ =1000 liters)

The length, width and height of Tank 1 is m meters each. 

Volume of Tank 1 = $$m^{3\ }$$ cubic meters

The length, width and height of Tank 2 is n meters each. 

Volume of Tank 2 = $$n^{3\ }$$ cubic meters

The length, width and height of Tank 3 is m, n and 1 m respectively. 

Volume of Tank 3 = mn cubic meters 

Since, Tank 1 is filled first and after Tank 1 is full, water is flowing into Tank 2 and Tank 3, 

Volume of water left in Tank 1 after Tank 2 and Tank 3 are filled = $$\left(m^{3\ }-n^3-mn\right)$$ cubic meters

Volume of water left in Tank 1 = 85000 L (given) = 85$$m^3$$

Hence, $$\left(m^{3\ }-n^3-mn\right)$$ = 85 , where, m and n are integers. 

$$n^3+mn=m^3-85$$

From option B, we get $$n^3+7n=258$$ $$\longrightarrow\ $$ n=6

From option C, we get $$n^3+5n=40$$ $$\longrightarrow\ $$ No integer solution

From option D, we get $$n^3+6n=131$$ $$\longrightarrow\ $$ No integer solution 

From option E, we get $$n^3+10n=915$$ $$\longrightarrow\ $$ No integer solution 

Hence, we got m = 7 and n = 6. 

$$\therefore\ $$ The required answer is B.