Problem 9 · 2011 Math Kangaroo
Medium
Number Theory
divisibilitycareful-counting
Andrew wrote down all the odd numbers from 1 to 2011 on a board. Bob then deleted all the multiples of three. How many numbers remained on the board?
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Answer: C — 671
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Hint 1 of 2
First count all odd numbers up to 2011, then remove the odd ones divisible by 3.
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Hint 2 of 2
Odd multiples of 3 are 3, 9, 15, … (every 6 apart).
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Approach: count odds, subtract odd multiples of three
- Odd numbers from 1 to 2011: (2011−1)/2 + 1 = 1006.
- Odd multiples of 3 are 3, 9, 15, …, 2007: there are 335 of them.
- Remaining = 1006 − 335 = 671.
Mark:
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