Problem 13 · 2014 Math Kangaroo
Hard
Number Theory
sum-constraint
A grandmother, her daughter and her granddaughter find that the sum of their ages is 100. Each age is a power of two (that is, several twos multiplied together). How old is the granddaughter?
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Answer: C — 4
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Hint 1 of 2
Each age is a power of two: 1, 2, 4, 8, 16, 32, 64, ...
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Hint 2 of 2
Find three powers of two that add to 100; the smallest one is the granddaughter.
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Approach: write 100 as a sum of three powers of two
- The grandmother must be the oldest power of two under 100, so try 64; that leaves 36 for the other two ages.
- Two powers of two adding to 36 can only be 32 + 4, so the ages are 64, 32 and 4.
- The youngest, the granddaughter, is 4.
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