Problem 27 · 2011 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
Seven years ago Eva’s age was a multiple of 8. In eight years it will be a multiple of 7. Eight years ago Raffi’s age was a multiple of 7. In seven years it will be a multiple of 8. Which of the following statements can be true?
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Answer: A — Raffi is two years older than Eva.
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Hint 1 of 2
Write each clue as 'age plus or minus something is a multiple of 7 or 8' and combine them.
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Hint 2 of 2
Compare the possible ages of Eva and Raffi to read off their age difference.
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Approach: find the smallest age each set of clues allows, then compare
- Eva needs \(E-7\) a multiple of 8 and \(E+8\) a multiple of 7; the smallest realistic age that fits both is \(E=55\) (since \(48\) and \(63\) work).
- Raffi needs \(R-8\) a multiple of 7 and \(R+7\) a multiple of 8; the smallest realistic age that fits both is \(R=57\) (since \(49\) and \(64\) work).
- With \(E=55\) and \(R=57\), Raffi is exactly 2 years older than Eva, so statement A can be true.
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