Question

Which of the following numbers is both a perfect square and a perfect cube?
i. 12 544
ii. 531 441
iii. 456 235
This is really stumping me up

Answers

  1. Answer:

    ii. 531441

    Step-by-step explanation:

    Find the prime factorization of each number.

    For a number to be a perfect square, it must have an even number of each factor.

    For a number to be a perfect cube, it must have a multiple of three of each factor.

    i. 12544 = 2^8 * 7^2

    There 8 factors of 2 and 2 factors of 7. The numbers of factors of both 2 and 7 are even, so it’s a perfect square, but neither 8 nor 2 is a multiple of 3, so it is not a perfect cube.

    ii. 531441 = 3^12

    12 is both even and a multiple of 3, so 531441 is a perfect square and a perfect cube.

    iii. 456235 = 5 * 13 * 7019

    No factor is a multiple of 2 or 3, so it is not a perfect square of perfect cube.

    Answer: ii. 531441

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