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Form the 999 sets of remainders modulo 1997, {0}, {i,1997-i}, for i=1,...,998. These sets partition all remainders. Hence two of the given integers, say x and y, have remainders which lie in one of these sets. If x and y have the same remainder, then they differ by a multiple of 1997. Otherwise, their remainders sum to 1997 and so x+y is a multiple of 1997.
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