Problem 27 · 2011 Math Kangaroo
Stretch
Number Theory
primescasework
For a positive whole number \(n \ge 2\), let \(\langle n\rangle\) denote the largest prime number less than or equal to n. How many positive whole numbers k satisfy the condition \(\langle k+1\rangle + \langle k+2\rangle = \langle 2k+3\rangle\)?
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Answer: B — 1
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Hint 1 of 2
Both sides are at most 2k+3, and the left side hits that maximum only when k+1 and k+2 are both prime.
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Hint 2 of 2
Test small k directly; the equality is very restrictive.
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Approach: bound both sides, then test small k
- The left side is at most (k+1)+(k+2) = 2k+3, and the right side ⟨2k+3⟩ is at most 2k+3.
- Equality forces a tight prime arrangement; checking k = 1 gives 2 + 3 = 5 = ⟨5⟩, which works.
- For k = 2, 3, 4, ... the largest primes drop below the needed totals, so none work.
- Exactly one k works, choice (B).
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