# Using prime factorization to solve it, but am still stuck?

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- micatkieLv 72 months agoFavorite Answer
1500k = 2² × 3 × 5³ × k

If 1500k is both a perfect square and a perfect cube,

the smallest value of 1500k = 2⁶ × 3⁶ × 5⁶

Hence, 2² × 3 × 5³ × (Smallest k) = 2⁶ × 3⁶ × 5⁶

Smallest k = 2⁴ × 3⁵ × 5³ = 486000

- MichaelLv 72 months ago
For a prime number to have a perfect square and perfect cube

the perfect number must be n⁶

n⁶ = (n²)³

n⁶ = (n³)²

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Prime factors of 1500

5³ * 2² * 3

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5⁶ = 5³ * 5³

2⁶ = 2² * 2⁴

3⁶ = 3 * 3⁵

Therefor

k = 5³ * 2⁴ * 3⁵

k = 486,000 <–––––––

- KrishnamurthyLv 72 months ago
Given that 1500k is both a perfect square and a perfect cube,

write down the smallest possible value of k.

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