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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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