Shnol D.E.

On the safe door, the expression $a^{n} b^{m} c^{k}$ is written. To open the door, each of the six letters must be replaced with a natural number so that the product becomes a cube of a natural number. Pinky, without thinking, has already replaced some three letters with numbers. Can Brain always replace the remaining three letters so that the door opens?
Let's consider two cases.

I. If Pinky replaced only one of the letters in each of the three powers, then Brain can simply make each of the three factors a perfect cube. Indeed, if in an expression of the form $x^{y}$ only one letter is replaced, then by choosing the remaining one, it is easy to make it a perfect cube (for this, one can either replace $x$ with a perfect cube or $y$ with a number divisible by three).

II. Suppose in one of the three factors, Pinky replaced both letters. Then in one of the factors, on the contrary, none of the letters were replaced. Let's say this is $c^{k}$. Then Brain can, for example, substitute the value of the expression $a^{n} b^{m}$ for $c$, and substitute two for $k$ - and the entire product will be equal to the cube of the number $a^{n} b^{m}$.

## Answer

Always.