A perfect square is defined as the square of an integer and

Question

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

don't know the OA

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

Bunuel · Accepted Answer

Given: positive integer n is a perfect square and a perfect cube --> n is of a form of n=x^6 for some positive integer x --> 0 0 x can be 1, 2 or 3 hence n can be 1^6, 2^6 or 3^6. Answer: B. n cannot be 0 as given that n is a positive integer .

Mackieman · Answer

Perfect cube: 
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 75
6^3 = 206
7^3 = 343
8^3 = 502
9^3 = 729
10^3 = 1000

If the square root of any of these numbers results in an integer then they will be both perfect square and perfect cube.

By looking at the numbers we can see that only three numbers results in integer, sqrt(1) =1, sqrt(4) = 2 and sqrt(9) = 3 => the answer is 3.

(ex sqrt(8^3) = sqrt(8*8*8) = sqrt(8)*sqrt(8)*sqrt(8) = 8*sqrt(8) = not integer)

amod243 · Answer

hi Mackieman

Agreed with your explanation. but i think we need to consider 0 also???

because 0 also satisfy the condition..

Mackieman · Answer

Well, if we see squares as the area of polygons

#
= area 1

##
##
= area 2*2 = 4

###
###
###
= area 3*3 = 9

In my opinion 0 shouldn't be considered to be a perfect square since 0 doesn't represent the area of square.

Bunuel · Answer

Zero is both perfect square and a perfect cube but  n  can not be 0 as given that  n  is a  positive integer .

144144 · Answer

cool question. thanks.

bankerboy30 · Answer

Bunuel - how did you come up with x^6?

Posted from my mobile device

vad3tha · Answer

n = a^2 = b ^3 ( with a, b, and n are positive integer) since a^2 = b^3 --> a^2 = (x^2)^3 = x^6 ( with b = x^2, if not a^2 cannot be equal b^3) or since b^3 = a^2 --> b^3 = x^6 ( same reason) --> n = a^2 = b^3 = x^6 Hope it helps a little bit :-D

Anyways, Thanks to Bunuel for his awesome explanation !!!! :-D

himanshujovi · Answer

Didn't get your explanation. 5^3 = 125 , 8^3 = 512 , 6^3 = 216.

IMO , the answer should be 2 as there are only two numbers which can come as perfect square as well as cube between 1 and 1000. The numbers would be 1 and 64

Bunuel · Answer

Check here: a-perfect-square-is-defined-as-the-square-of-an-integer-and-108103.html#p856696

praveenboddu · Answer

After reading the question it will be a bit scary. But there is <1 min logic for this question.

For a number to be a square and cube at the same time, it should be in the power of 6. X^6. We have to Start counting from 1

1^6 = 1
2^6 = 64
3^6 = 729
4^6 will be greater than 1000. So the Answer is 3

JeffTargetTestPrep · Answer

We note that if an integer is a perfect square and a perfect cube at the same time, then it is the sixth power of some integer. We need to determine how many numbers exist that when raised to the 6th power are less than 1,000. 1^6 = 1 < 1000 2^6 = 64 < 1000 3^6 = 729 < 1000 4^6 = 4,096 > 1000 Since 4^6 > 1000, we see that only 3 values exist. Answer: B

PranayBudime · Answer

To Solve this problem, we need to know that for a number to be perfect square it can be expressed in powers of 2.

1. Cubes which are less than 1000 are  1^3,2^3,3^3,4^3,5^3,6^3,7^3,8^3,9^3  
2. Out of the above cubes, only  1^3, 4^3,9^3  can be expressed in even powers (i.e.  1^3=1=1^2 ,  4^3=2^6  and  9^3=3^6 )

So only 3 numbers satisfy the condition

andr3 · Answer

Bullseye, as always! Thank you Bunuel! Your help is always highly appreciated!

3176401 · Answer

Where is this question taken from?

prashu1701 · Answer

Easy question not 700+ and don't waste time in such question ... Approach should be...

u need to find cubes which are also expressed as square.(1^3,2^3,3^3......10^3)
Since 10^3 = 1000. Don't go beyond that.
1 to 10 we have 3 sqaures 1,4,9, so answer is 3.
Coz
4^3 = 2^(2*3) = 8^2.
9^3 = 3^(2*3) = 27^2.
Similarly 1.
Hope this helps...don't start sloving questions as soon as u read.understand the language, most of the questions doesn't require calculations

Posted from my mobile device

GMATinsight · Answer

Q-49
A perfect square must be of the form a^2
A perfect Cube must be of the form b^3

Therefore a number which is a perfect square as well as a perfect cube must have exponent multiple of 2 and 3 both i.e. Multiple of 6

Numbers of our interest will be of the form p^6

1^6 = 1
2^6 = 64
3^6 = 729
4^6 = 4096
i.e. we have only three such numbers from 1 to 1000

ANswer: Option B

hellypatel · Answer

1 
64 - perfect square of 8 and perfect cube of 4
729 - perfect square of 27 and perfect cube of 9

so B

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A perfect square is defined as the square of an integer and

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