Problem 16 · 2023 Math Kangaroo
Medium
Number Theory
divisibilityoff-by-one
Four posts are placed along a 120 m long running track at the distances shown. How many more posts must be added so that the track is divided into sections that are all the same length?
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Answer: C — 17
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Hint 1 of 2
The existing posts sit at distances 24 m, 54 m and 120 m from the start; the equal sections must line up with all of them.
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Hint 2 of 2
Find the largest length that divides all those distances, then count how many posts that many sections need.
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Approach: use the greatest common divisor of the existing post positions
- Measured from the start, the four posts are at 0, 24, 54 and 120 metres.
- Equal sections must reach each existing post, so the section length must divide 24, 54 and 120; the largest such length is their greatest common divisor, 6 m.
- That makes 120 / 6 = 20 sections, which need 21 posts in all.
- Since 4 posts are already there, 21 − 4 = 17 more are needed, answer C.
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