The question is from a book used for transition between high school mathematics and university mathematics, which states: Prove the following statement or give a counterexample
$\forall n \in \mathbb Z, \exists a,b,c,d,e,f,g,h \in \mathbb Z$ such that $n=a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3$
After several trials I believe the statement is true but find it difficult to prove. Could somebody offer a solution without reference to congruence? I can see that if one choose $a,b,c,d,e,f,g$ equals from $0 \:to\: 1$ each then $n \in {1,2,3,4,5,6,7,8} $ can be represented. For the next eight numbers, by choosing $a=2$ and adjusting the remaining parameters from $0\:to\:1$, can also be achieved. Thus I suspect all numbers can be represented this way, but, having said that, I still feel hard to find a proof.