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# A perfect square and A perfect cube

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Manager
Joined: 19 Feb 2009
Posts: 58
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Kudos [?]: 13 [0], given: 8

A perfect square and A perfect cube [#permalink]  22 Jan 2011, 03:49
00:00

Difficulty:

5% (low)

Question Stats:

33% (01:47) correct 66% (01:06) wrong based on 15 sessions
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
[Reveal] Spoiler: OA

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Working without expecting fruit helps in mastering the art of doing fault-free action !

Manager
Joined: 20 Dec 2010
Posts: 168
Location: Stockholm, Sweden
Followers: 2

Kudos [?]: 26 [0], given: 5

Re: A perfect square and A perfect cube [#permalink]  22 Jan 2011, 04:39
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)
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Manager
Joined: 19 Feb 2009
Posts: 58
Followers: 2

Kudos [?]: 13 [0], given: 8

Re: A perfect square and A perfect cube [#permalink]  22 Jan 2011, 07:25
hi Mackieman

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

because 0 also satisfy the condition..
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Working without expecting fruit helps in mastering the art of doing fault-free action !

Math Expert
Joined: 02 Sep 2009
Posts: 16790
Followers: 2770

Kudos [?]: 17572 [0], given: 2184

Re: A perfect square and A perfect cube [#permalink]  22 Jan 2011, 08:12
Expert's post
amod243 wrote:
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

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<x^6<10^3 --> 0<x^2<10 --> x can be 1, 2 or 3 hence n can be 1^6, 2^6 or 3^6.

amod243 wrote:
hi Mackieman

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

because 0 also satisfy the condition..

n can not be 0 as given that n is a positive integer.
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Manager
Joined: 20 Dec 2010
Posts: 168
Location: Stockholm, Sweden
Followers: 2

Kudos [?]: 26 [0], given: 5

Re: A perfect square and A perfect cube [#permalink]  22 Jan 2011, 08:19
amod243 wrote:
hi Mackieman

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

because 0 also satisfy the condition..

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.
_________________

12/2010 GMATPrep 1 620 (Q34/V41)
01/2011 GMATPrep 2 640 (Q42/V36)
01/2011 GMATPrep 3 700 (Q47/V39)
02/2011 GMATPrep 4 710 (Q48/V39)
02/2011 MGMAT CAT 1 650 (Q46/V32)
02/2011 MGMAT CAT 2 680 (Q46/V36)
02/2011 MGMAT CAT 3 710 (Q45/V41)

Math Expert
Joined: 02 Sep 2009
Posts: 16790
Followers: 2770

Kudos [?]: 17572 [0], given: 2184

Re: A perfect square and A perfect cube [#permalink]  22 Jan 2011, 08:25
Expert's post
Mackieman wrote:
amod243 wrote:
hi Mackieman

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

because 0 also satisfy the condition..

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.

Zero is both perfect square and a perfect cube but n can not be 0 as given that n is a positive integer.
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Senior Manager
Joined: 08 Nov 2010
Posts: 424
Followers: 6

Kudos [?]: 28 [0], given: 161

Re: A perfect square and A perfect cube [#permalink]  18 Feb 2011, 13:59
cool question. thanks.
Re: A perfect square and A perfect cube   [#permalink] 18 Feb 2011, 13:59
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