Problem 12 · 2018 Math Kangaroo
Hard
Number Theory
divisibilitycareful-counting
105 numbers are written in a row: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, … where each number n is written exactly n times. How many of those numbers are divisible by 3?
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Answer: D — 30
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Hint 1 of 2
1 + 2 + 3 + … + 14 = 105, so the row runs from 1 up to 14.
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Hint 2 of 2
Only the multiples of 3 (namely 3, 6, 9, 12) count, and each appears as many times as its value.
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Approach: sum the counts of the multiples of 3 up to 14
- Since 1 + 2 + … + 14 = 105, the numbers used run from 1 to 14.
- The multiples of 3 in that range are 3, 6, 9 and 12, each written that many times.
- Total = 3 + 6 + 9 + 12 = 30.
Mark:
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