2002 IMO Shortlist Problems/N1

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Problem

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with

$x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}$?

Solution

Observe that $2002^{2002}\equiv 4^{2002}\equiv 64^{667}\cdot 4\equiv 4\pmod{9}$. On the other hand, each cube is congruent to 0, 1, or -1 modulo 9. So a sum of at most three cubes modulo 9 must among $0,\pm 1,\pm 2,\pm 3$ none of which are congruent to 4. Therefore $t\geq 4$.


To show that 4 is the minimum value of $t$, note that $(10\cdot 2002^{667})^3+(10\cdot 2002^{667})^3+(2002^{667})^3+(2002^{667})^3=2002^{2002}$

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