Problem 7 · Track prime exponents · 5 points
Made by AI and not reviewed by a real person.
N is the smallest positive integer such that N ÷ 2 is a perfect square and N ÷ 3 is a perfect cube. What is N?
- 648
- 324
- 216
- 108
- 432
Show a hint
- Write N = 2ᵃ·3ᵇ and translate each condition into conditions on a and b.
- N/2 square → a−1 even, b even. N/3 cube → a divisible by 3, b−1 divisible by 3.
Show the answer and solution
Answer: A — 648
Let N = 2ᵃ3ᵇ. N/2 = 2ᵃ⁻¹3ᵇ must be a square → a−1 and b even. N/3 = 2ᵃ3ᵇ⁻¹ must be a cube → a and b−1 divisible by 3. Smallest a: odd and divisible by 3 → 3. Smallest b: even with b−1 divisible by 3 → 4. N = 2³·3⁴ = 8 · 81 = 648 (check: 648/2 = 324 = 18², 648/3 = 216 = 6³). No other prime can help: its exponent would have to be even AND divisible by 3, and the smallest such exponent, 0, means "leave it out". Translating a number problem into bookkeeping on prime exponents is a signature olympiad technique.