Problem 23 · 2025 Math Kangaroo
Stretch
Number Theory
number-systemsfactorization
What is the smallest positive integer N such that the expression \(\sqrt{2\sqrt{3\sqrt{N}}}\) has an integer value?
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Answer: E — another number
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Hint 1 of 2
Write the nested radical as a single product of fractional powers of 2 and 3.
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Hint 2 of 2
The value is 21/2·31/4·N1/8; choose N = 2a3b to clear all the fractions.
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Approach: combine the radicals into fractional exponents
- √(2√(3√N)) = 21/2·31/4·N1/8.
- Need (4 + a)/8 and (2 + b)/8 to be integers, so smallest a = 4, b = 6: N = 2⁴·3⁶ = 11664.
- That is none of A–D, so the answer is another number.
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