# A big cube and 99 smaller cubes

A big cube is cut into 99 smaller cubes. Exactly 98 of these 99 smaller cubes are unit cubes.

Question: What is the volume of the big cube?

Hm...

Let $k$ denote the side length of the non-unit cube. We know $k>1$ because if it weren't, then there couldn't be a cube, and for the same reason we know $k\in\mathbb{N}$.

Let $n$ denote the length of the full cube. This means $n^3-k^3=98$.

That factors into $(n-k)(n^2+nk+k^2)=98$; the only factors of 98 that produce an integer value for $n$ and $k$ are 2 and 49. This makes $n=5$ and $k=3$.

So the volume of the full cube is 125 cubic units.

• You can factor $n^3-k^3$ as $(n-k)(n^2+nk+k^2)$ and then try each of the possibilities for factoring 98. Only $2\times49$ provides an integer solution. – f'' Oct 18 '15 at 15:52
• @f'': Right, I always forget about the difference of cubes factorization. That would definitely make it easier. – Deusovi Oct 18 '15 at 16:03