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999 distinct remainders with one of them zero

Source: XIX Olimpíada Matemática Rioplatense (2010)

July 22, 2011

modular arithmeticnumber theory unsolvednumber theory

Problem Statement

Let

r_2, r_3,\ldots, r_{1000}

denote the remainders when a positive odd integer is divided by

2,3,\ldots,1000

, respectively. It is known that the remainders are pairwise distinct and one of them is

0

. Find all values of

k

for which it is possible that

r_k = 0