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
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Find all n such that 0 < n^6 < 1000
ReplyDelete10^3 = (greater than 3)^3
1,2, 3 are the possible ones.
(B)
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.
ReplyDeleteThose 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.
If you look closely you can generalize the numbers as
ReplyDelete1 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.
Why have you not considered 0? I think including 0, the answer should be C) or 4.
ReplyDeletethe question asks positive integers zero is nether positive nor negative
ReplyDeletethe number are 1, 64 and 729.