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Multiplications and digits

Source: Polish Mathematical Olympiad Second Round (day 2)

February 23, 2008
number theoryrelatively primenumber theory proposed

Problem Statement

We have a positive integer n n such that n≠3k n \neq 3k. Prove that there exists a positive integer m m such that ∀k∈N k≥m k \forall_{k\in N \ k\geq m} \ k can be represented as a sum of digits of some multiplication of n n.
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