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

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  • 2 months ago
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    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

  • 2 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 

    -----------------------

    Prime factors of 1500

    5³ * 2²  * 3

    -----------------------

    5⁶ = 5³ * 5³ 

    2⁶ = 2² * 2⁴ 

    3⁶ = 3 * 3⁵ 

    Therefor 

    k = 5³ * 2⁴ * 3⁵ 

    k = 486,000 <–––––––

  • 2 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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