Problem 21 · 2022 Math Kangaroo
Stretch
Number Theory
sum-constraintdivisibility
Once I met six sisters whose ages were six consecutive integers. I asked each one of them: How old is the oldest of your sisters? Which of the following numbers cannot be the sum of the six answers?
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Answer: D — 205
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Hint 1 of 2
Each sister names her oldest sister: five of them name the same person, but the oldest names someone different.
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Hint 2 of 2
Write the total of the six answers as one expression in the youngest age, then check it against each option.
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Approach: build the sum of the six answers, then test the fixed remainder
- Call the ages \(n, n+1, \dots, n+5\); the oldest is \(n+5\), and her own oldest sister is \(n+4\).
- Five sisters answer \(n+5\) and the oldest answers \(n+4\), so the six answers add to \(5(n+5)+(n+4)=6n+29\).
- So any valid sum minus 29 must be a multiple of 6; checking the options, only \(205-29=176\) is not, so 205 cannot occur.
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