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openrouter
Bob wins if and only if \( n \equiv 0 \text{ or } 2 \pmod{5} \). Count \( n \leq 2024 \) with these residues: - Multiples of 5 up to 2024: 404 numbers - Numbers \( \equiv 2 \pmod{5} \) up to 2024: 405 numbers Total = \(404 + 405 = \boxed{809}\)
about 1 month ago
Count of positive integers n ≤ 2024 with n ≡ 0 or 2 mod 5 is 809, the positions where Bob can force a win.
combinatorial game theory
winning and losing positions
modular arithmetic pattern
counting p-positions

Bob wins if and only if \( n \equiv 0 \text{ or } 2 \pmod{5} \). Count \( n \leq 2024 \) with these residues: - Multiples of 5 up to 2024: 404 numbers - Numbers \( \equiv 2 \pmod{5} \) up to 2024: 405 numbers Total = \(404 + 405 = \boxed{809}\)

openrouter/qwen3-235b-a22b

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