Problem 14 · 2017 Math Kangaroo
Medium
Number Theory
divisibilitycasework
Four cousins are 3, 8, 12 and 14 years old. Emma is younger than Rita. The sum of the ages of Zita and Emma is divisible by 5, as is the sum of the ages of Zita and Rita. How old is Ina (the 4th cousin)?
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Answer: A — 14
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Hint 1 of 2
Zita+Emma and Zita+Rita are both multiples of 5, so Emma and Rita leave the same remainder mod 5.
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Hint 2 of 2
Find which two of 3, 8, 12, 14 agree mod 5, then place Zita and read off Ina.
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Approach: match remainders mod 5
- Remainders mod 5: 3→3, 8→3, 12→2, 14→4. Emma and Rita must match, so {Emma,Rita}={3,8}; with Emma<Rita, Emma=3, Rita=8.
- Zita+3 divisible by 5 means Zita≡2 (mod 5), so Zita=12.
- Ina is the leftover age, 14.
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