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Problem of the Month – October 2021

An integer number s is said to be n-smooth if s = a 2 1 + a 2 2 + · · · + a 2 n , where each ai , i = 1, 2, . . . , n is divisible by n. An integer number s is said to be n-rough if s = a 2 1+a 2 2+· · ·+a 2 n , where each ai , i = 1, 2, . . . , n is not divisible by n. Find all positive integers n for which any n-smooth number is n-rough number.