Some help/hint would be greatly appreciated :)
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Codeforces (c) Copyright 2010-2025 Mike Mirzayanov
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There are at most 11 prime numbers less than sqrt(N). Try all possible assignments of these switches (211 ways). Each lamp is now attached to at most 1 unassigned switch (those are greater than sqrt(N)) and we can solve it by greedy approach (count the number of on/off lamps for each switch).