<s>[INST] Smallest positive integer that has $x_{1},x_{2}, \dots, x_{t}$ with $x_1^3 + x_2^3 + \dots + x_t^3 = 2002^{2002}?$ 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. 
 [/INST] 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 $t$.
</s>