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## Sufficient condition for a sequence to be an enumeration of integers

Let be a sequence if integers with infinitely many positive and negative terms. Suppose that for any the numbers have different remainders modulo . Prove that each integer occurs exactly once in the given sequence.

*IMO 2005 Problem 2*

Consider . Suppose that for some . Then in ; Contradiction. Therefore for every .

Consider the greatest element of . Then, since are distinct, it follows that must be a list of consecutive integers.

Since the initial sequence contains infinitely many positive and negative integers it follows that is an enumeration of the integers.

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Categories: Algebra, IMO, Number theory, Olympiad, Problem Solving
IMO, TST

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