The figures below show a way to cut a cube into smaller cubes.
![](https://cdn.mathpix.com/cropped/2024_05_01_ffb48c2069f209c1311cg-19.jpg?height=324&width=1142&top_left_y=1408&top_left_x=526)

Notice that we start with a single cube, and after the cuts, we end up with 8 smaller cubes. If we choose one of these cubes and cut it into 8 smaller cubes, we will end up with exactly 15 smaller cubes. Observe that these smaller cubes are not all the same. Suppose this process is repeated several times. The questions are:

(a) How many times must we perform this process to obtain exactly 99 smaller cubes?

(b) By repeating this process, is it possible to obtain exactly 2016 smaller cubes?
Solution

(a) Observe that at each step we have a surplus of 7 in the number of cubes. Therefore, after repeating the process $x$ times, we will have exactly $7 x+1$ cubes. Consequently, to find out the number of steps required to obtain 99 cubes, we just need to solve the equation:

$$
\begin{aligned}
7 x+1 & =99 \\
7 x & =98 \\
x & =14
\end{aligned}
$$

Thus, to obtain exactly 99 cubes, we need to perform the process 14 times.

(b) Knowing that after repeating the process $x$ times we have $7 x+1$ cubes, we can conclude that the quantities of cubes obtained must leave a remainder of 1 when divided by 7. Since 2016 leaves a remainder of 0 when divided by 7, this quantity of cubes cannot be obtained.