CF492E Vanya and Field

Vanya decided to walk in the field of size $ n×n $ cells. The field contains $ m $ apple trees, the $ i $ -th apple tree is at the cell with coordinates $ (x_{i},y_{i}) $ . Vanya moves towards vector $ (dx,dy) $ . That means that if Vanya is now at the cell $ (x,y) $ , then in a second he will be at cell . The following condition is satisfied for the vector: , where  is the largest integer that divides both $ a $ and $ b $ . Vanya ends his path when he reaches the square he has already visited. Vanya wonders, from what square of the field he should start his path to see as many apple trees as possible.

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In the first sample Vanya's path will look like: $ (1,3)-(3,1)-(0,4)-(2,2)-(4,0)-(1,3) $ In the second sample: $ (0,0)-(1,1)-(0,0) $