Problem 24 · 2016 Math Kangaroo
Stretch
Number Theory
divisibility
Little Red Riding Hood is taking waffles to three grandmothers. Her basket starts completely full. Just before she reaches each grandmother’s house, the wolf eats half of the waffles in the basket. When she leaves the third grandmother, the basket is empty. Each grandmother gets the same number of waffles. The original number of waffles can definitely be divided by which of the following numbers?
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Answer: D — 7
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Hint 1 of 2
Work backward: before each grandmother the basket held twice what it held after.
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Hint 2 of 2
Each grandmother gets the same amount; reconstruct the start as a multiple, then see what must divide it.
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Approach: work backward through the halvings
- The wolf halves the basket before each of the 3 grandmothers, and each grandmother takes the same amount g.
- Working backward through the three halvings and equal gifts, the original amount comes out as a multiple of 7.
- So the original number of waffles is always divisible by 7.
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