Monday, May 4, 2009

Problem 24 - GMAT Powers & Roots

A perfect square is defined as the square of an integer and a perfect cube is defined as the cube of an integer. How many positive integers n are there such that n is less than 1,000 and at the same time n is a perfect square and a perfect cube?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

5 comments:

  1. Find all n such that 0 < n^6 < 1000

    10^3 = (greater than 3)^3

    1,2, 3 are the possible ones.

    (B)

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  2. I think the answer should be A because between 1 and 1000 we only have two integers which are perfect square and perfect cube at the same time.
    Those integers are: 1 - sqrt(1) =cubert(1) =1 integer
    second integer is 64 - sqrt(64) =8 cubert(64) = 4 they both are integers.
    Hence A.

    ReplyDelete
  3. If you look closely you can generalize the numbers as
    1 and
    (2k)^2 x (2k)^2 x (2k)^2= 64K^6 and
    (3k)^2 x (3k)^2 x (3k)^2 = 729k^6
    answer is 3. Or as Blaoism suggested:
    0 < n^6 < 10^3 or
    n^2 < 10 (valid for 1,2,3)
    Hence answer is 3.

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  4. Why have you not considered 0? I think including 0, the answer should be C) or 4.

    ReplyDelete
  5. the question asks positive integers zero is nether positive nor negative
    the number are 1, 64 and 729.

    ReplyDelete