XXXI Brazilian Math Olympiad

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 $n_0=1+7\cdot 26-7-26+1$.