At the first quadrant in the -plane, you are asked to construct a line segment with the following specifications:
  1. Select a prime point along the x-axis.
  2. Select an integer point along the y-axis.
  3. Connect these points to form a line segment.
How many line segments, with the abovementioned specifications, can be constructed that pass through a given point ?
For example, in the figure below, it is shown that 2 such lines pass through the point , with lines connecting prime points and to integer points and , respectively. It can be proven that these are the only line segments, drawn as specified, that pass through .
NOTE: It is possible that no line segments, as specified, pass through a given point.
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Last Solution submitted on Nov 28, 2025

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David Hill Ramon Villamangca GeeTwo Mehdi Dehghan

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