2002 IMO Shortlist Problems/N1
Problem
What is the smallest positive integer such that there exist integers
with
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Solution
Observe that . 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
none of which are congruent to 4. Therefore
.
To show that 4 is the minimum value of , note that