What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \dots, x_{t}$ with $x_1^3 + x_2^3 + \dots + x_t^3 = 2002^{2002}?$
I don't really get what the question means by "such that there exist integers $x_{1},x_{2}, \dots, x_{t}$ " and I don't know how to start.
Best Answer
Evaluating $2002^{2002}$ mod $9$, we get \begin{align*} 2002^{2002} \!&\equiv 4^{2002}\;(\text{mod}\;9)\\[4pt] \!&\equiv 4{\,\cdot\,}4^{2001}\;(\text{mod}\;9)\\[4pt] \!&\equiv 4{\,\cdot\,}(4^3)^{667}\;(\text{mod}\;9)\\[4pt] \!&\equiv 4{\,\cdot\,}1^{667}\;(\text{mod}\;9)\\[4pt] \!&\equiv 4\;(\text{mod}\;9)\\[4pt] \end{align*} Noting that $\{x^3\;\text{mod}\;9\mid 0\le x \le 8\}=\{0,1,8\}$, it's easily verified that $$x_1^3+x_2^3+x_3^3\equiv 4\;(\text{mod}\;9)$$ has no integer solutions.
It follows that the equation $$x_1^3+x_2^3+x_3^3=2002^{2002}$$ has no integer solutions.
Hence the least qualifying value of $t$ must be greater than $3$.
But identically we have \begin{align*} 2002^{2002} &= 2002^{2001}{\,\cdot\,}\,2002\\[4pt] &= 2002^{2001}{\,\cdot\,}(1000+1000+1+1)\\[4pt] &= \left(2002^{667}\right)^3{\,\cdot\,}(10^3+10^3+1^3+1^3)\\[4pt] \end{align*} which is a sum of $4$ cubes.
Therefore $t=4$ is the least qualifying value of $t$.