Problem 9 · 2016 Math Kangaroo
Easy
Number Theory
divisibility
Alex has one rope 1 m long and another 2 m long. He cuts up both ropes so that all the pieces are of equal length. Which of the following numbers of pieces can he not obtain this way?
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Answer: B — 8
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Hint 1 of 2
Every piece must fit a whole number of times into the 1 m rope and into the 2 m rope.
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Hint 2 of 2
If each piece is 1/n of a metre, the short rope gives n pieces and the long rope gives 2n, so the total is always a multiple of 3.
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Approach: the total number of pieces is always a multiple of 3
- Equal pieces must divide both ropes exactly, so each piece is 1/n of a metre for some whole number n.
- Then the 1 m rope makes n pieces and the 2 m rope makes 2n pieces, for 3n pieces in all — always a multiple of 3.
- Among the options 6, 9, 12 and 15 are multiples of 3, but 8 is not, so 8 pieces cannot be obtained.
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