Problem 16 · 2015 Math Kangaroo
Medium
Number Theory
caseworksum-constraint
A bush has 10 twigs. Each twig has exactly 5 leaves, or exactly 2 leaves and a flower. Which of the following numbers could be the total number of leaves on the bush?
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Answer: E — None of the numbers from (A) to (D).
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Hint 1 of 2
Each of the 10 twigs adds either 5 leaves or 2 leaves.
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Hint 2 of 2
Write the total as \(5a + 2b\) with \(a + b = 10\) and check which listed number is reachable.
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Approach: reachable totals from 5s and 2s on 10 twigs
- Each twig gives 5 leaves or 2 leaves, so with 10 twigs the total is \(5a + 2b\) where \(a + b = 10\).
- Then the total is \(2 \cdot 10 + 3a = 20 + 3a\), so it is 20 more than a multiple of 3.
- The reachable totals run 20, 23, 26, …, 50; none of 45, 39, 37, 31 fits this pattern.
- So the answer is None of the numbers (E).
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