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American Remainder Theorem

Source: 2017 AIME I #2

March 8, 2017
AMCAIME2017 AIME I

Problem Statement

When each of 702, 787, and 855 is divided by the positive integer mm, the remainder is always the positive integer rr. When each of 412, 722, and 815 is divided by the positive integer nn, the remainder is always the positive integer s≠rs \neq r. Fine m+n+r+sm+n+r+s.
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