Problem 23 · 2016 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
Leo has built a stick made up of 27 building blocks (see picture). He splits the stick into two pieces so that one part is twice as long as the other. He keeps repeating this: each time he takes one of the two pieces and splits it so that one piece is twice as long as the other. Which of the following pieces can never result in this way? (The choices are pieces of length 2, 4, 6, 8 and 10 blocks.)
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Answer: E — 10
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Hint 1 of 2
Splitting a piece into a 2:1 ratio only works when its length divides into three equal parts.
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Hint 2 of 2
List every length you can reach starting from 27 and see which option never appears.
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Approach: track which lengths a 2:1 split can produce
- 27 splits into 18 and 9; 18 splits into 12 and 6; 9 splits into 6 and 3; 12 splits into 8 and 4; 6 splits into 4 and 2.
- The reachable lengths are 2, 3, 4, 6, 8, 9, 12, 18, 27.
- A 10-block piece never appears, so 10 can never result.
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