Consider the following two operations, A and B: A consists in cutting a cube in 8 equal cubes of half its dimensions; B consists in cutting a cube in 27 equal cubes of one third of its dimensions. A and B increases the total number of cubes in 7 and 26, respectively, so one can obtain $1 + 7x + 26y$ cubes by performing A $x$ times and B $y$ times. Since $\text{gcd}(7, 26) = 1$ and $x, y$ are non-negative integers, one can obtain any number of cubes bigger than $1 + 7 \cdot 26 - 7 - 26$, so one can choose $n_0 = 1 + 7 \cdot 26 - 7 - 26 + 1$.