Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?
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Tags: algebra, number theory
miyukina
26.03.2024 23:22
Column can take 0 or 5 while row can take 0, 3 or 6 We’re already shown that R1, R2, C1 and C2 are nonzeros, so C1 = C2 = 5 Because both R1 and R2 need at least two more nonzero columns, then we know that there are at least 10 + 2 × 5 = 20 checkers That means our count of checkers, d, is at least 20 and at most 40 (when all columns take in 5 so 8 × 5 = 40) but d must also follow the rules about the rows 3d With 20 ≤ d ≤ 40 and d = 0 mod 3 Answer = 30