<s>[INST] Show that, for any two integers $a$ and $b$, at least one of the expressions $a^3, b^3, a^3+b^3$ or $a^3-b^3$ will be divisible by $7$. This is a very interesting word problem that I came across in an old textbook of mine:

Show that, for any two integers $a$ and $b$, at least one of the expressions $a^3, b^3, a^3+b^3$ or $a^3-b^3$ will be divisible by $7$.

So I know it's got something to do with Euler's totient function and the Euler-Fermat theorem, which yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance.
 [/INST] Hint:
Note that, $a^3\equiv 0,1^3,2^3,3^3,4^3,5^3,6^3(\mod 7)\Rightarrow a^3\equiv 0,1,1,-1,1,-1,-1(\mod 7)$

More explanation :
Then $7\mid a\Rightarrow 7\mid a^3$ or $7\mid b\Rightarrow 7\mid b^3$.
If $7$ divides none of $a,b$, then either $a^3+b^3\equiv 0(\mod 7)$ or $a^3-b^3\equiv 0(\mod 7)$.
</s>