Problem 21 · 2011 Math Kangaroo
Stretch
Number Theory
divisibility
A drum contains a number of balls, each marked with a different positive whole number. On 30 of the balls the numbers are divisible by 6, on 20 balls the numbers are divisible by 7, and on 10 balls the numbers are divisible by 42. What is the minimum number of balls in the drum?
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Answer: B — 40
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Hint 1 of 2
Balls labelled with a multiple of 42 are counted in both the 'divisible by 6' and 'divisible by 7' groups.
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Hint 2 of 2
Use inclusion–exclusion on the 6- and 7-divisible counts.
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Approach: inclusion–exclusion on overlapping divisibility groups
- Multiples of 42 are exactly the numbers divisible by both 6 and 7, and there are 10 of them.
- Balls divisible by 6 or 7 = 30 + 20 − 10 = 40.
- The drum needs at least 40 balls.
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