A gardener plants 1000 trees in a square grid with equal rows and columns. How many trees remain after forming the largest possible square ?
39
A square grid with equal rows and columns holds a perfect square number of trees, so the largest square uses the largest perfect square not exceeding 1000.
\(31^{2} = 961\) and \(32^{2} = 1024 \gt 1000\), so the largest square has 31 rows and 31 columns.
Trees left over \(= 1000 - 961 = 39\).
Check: \(961 + 39 = 1000\).
Hence, 39.
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