The question: "Prove that there are no positive perfect cubes less than 1000 that are the sum of the cubes of two positive integers." How would you go about picking a method to prove this? Is there a better way to do it other than by exhaustion?
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There are only $9$ perfect cubes less than $1000$, so there are not many cases to check. It is true that showing there are no solutions to $x^3+y^3=z^3$ is the easiest case of Fermat's last theorem, but it still seems harder than exhaustion here. Both will work. The second is more useful going forward.
The only cubic residues mod 7 are 0 and $\pm 1$. So if $a^3+b^3=c^3$, either
That's 3+9=12 cases to check by exhaustion, which is not so bad.