Problem 8 · 2015 Math Kangaroo
Easy
Number Theory
perfect-squarefactorization
Which of the following numbers is neither a square nor a cubic number?
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Answer: A — \(6^{13}\)
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Hint 1 of 2
A power is a perfect square when its exponent is even, and a perfect cube when the exponent is a multiple of 3.
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Hint 2 of 2
Check each exponent for divisibility by 2 or by 3; the odd-and-not-multiple-of-3 one is neither.
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Approach: test the exponent's divisibility
- 6¹³: exponent 13 is neither even nor a multiple of 3 → neither a square nor a cube.
- 5¹² (12), 4¹¹ = 2²² (22), 3¹⁰ (10) are all squares; 2⁹ (9) is a cube.
- So the odd one out is 6¹³.
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