Problem 17 · 2017 Math Kangaroo
Medium
Number Theory
off-by-onecareful-counting
Simon wants to cut a piece of wire into 9 equally long pieces and makes marks where he needs to make his cuts. Barbara wants to cut the same piece of wire into 8 equally long pieces and makes marks where she needs to make her cuts. Carl cuts the piece of wire at every mark. How many pieces does Carl get?
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Answer: B — 16
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Hint 1 of 2
Count the cut marks each person makes, then check whether any marks land on the same spot.
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Hint 2 of 2
Marks at ninths and eighths of the wire never coincide in the interior.
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Approach: count distinct marks, add one for pieces
- Nine equal pieces need 8 interior marks; eight equal pieces need 7 interior marks.
- Since 8 and 9 share no common factor, no ninth-mark equals an eighth-mark, giving 8 + 7 = 15 distinct marks.
- 15 cuts make 15 + 1 = 16 pieces.
Mark:
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