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?
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1 Answer
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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.
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4$\begingroup$ 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. $\endgroup$– f''Oct 18, 2015 at 15:52
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$\begingroup$ @f'': Right, I always forget about the difference of cubes factorization. That would definitely make it easier. $\endgroup$– Deusovi ♦Oct 18, 2015 at 16:03