IMC 2014 Q18
18. Consider looking from the origin (0, 0) towards all the points (\(m\), \(n\)), where each of \(m\) and \(n\) is an integer. Some points are hidden, because they are directly in line with another nearer point. For example, (2, 2) is hidden by (1, 1).
How many of the points (6, 2), (6, 3), (6, 4), (6, 5) are not hidden points?
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2014, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 1
The point (\(m\), \(n\)) is hidden if and only if \(m\) and \(n\) share a common factor greater than 1. So (6, 2) is hidden by (3, 1) since 6 and 2 have common factor 2. Also (6, 3) is hidden by (2, 1) whilst (6, 4) is hidden by (3,2). However, 6 and 5 have no common factor other than 1 and therefore (6, 5) is not a hidden point.
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).