Problem 12 · 2013 Math Kangaroo
Medium
Number Theory
divisibility
How many positive integers n are there for which both n⁄3 and \(3n\) are three-digit numbers?
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Answer: A — 12
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Hint 1 of 2
Write both 'three-digit' conditions as ranges for n.
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Hint 2 of 2
Then keep only the n that are multiples of 3.
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Approach: intersect two ranges
- n/3 three-digit means 300 ≤ n ≤ 2997; 3n three-digit means 34 ≤ n ≤ 333.
- Together: 300 ≤ n ≤ 333 with n a multiple of 3.
- Those are 300,303,…,333 — twelve values, so A.
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