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It is well known that, in an exact covering system in ℤ, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic number field. In this paper, we prove Kim's conjecture. In other words, we prove that exact covering systems in any algebraic number field must have repeated moduli.

EXACT COVERING SYSTEMS IN NUMBER FIELDS