A perfect square is defined as the square of an integer and : Problem Solving (PS)

amod243

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A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 04:49

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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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Bunuel

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A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 09:12

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Expert Reply

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.

Answer: B.

amod243 wrote:

hi Mackieman

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

because 0 also satisfy the condition..

n cannot be 0 as given that n is a positive integer.
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Mackieman

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Re: A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 05:39

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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

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Joined: 19 Feb 2009

Posts: 32

Schools:INSEAD,Nanyang Business school, CBS,

Re: A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 08:25

hi Mackieman

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

because 0 also satisfy the condition..

Mackieman

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Joined: 20 Dec 2010

Posts: 128

Location: Stockholm, Sweden

Re: A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 09:19

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.

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Bunuel

Math Expert

Joined: 02 Sep 2009

Posts: 92747

Re: A perfect square is defined as the square of an integer and [#permalink]

22 Jan 2011, 09:25

Expert Reply

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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Re: A perfect square is defined as the square of an integer and [#permalink]

18 Feb 2011, 14:59

cool question. thanks.

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bankerboy30

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Joined: 27 May 2014

Posts: 72

Re: A perfect square is defined as the square of an integer and [#permalink]

19 Jul 2014, 17:21

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

Posted from my mobile device

vad3tha

Manager

Joined: 22 Feb 2009

Posts: 109

A perfect square is defined as the square of an integer and [#permalink]

19 Jul 2014, 23:52

bankerboy30 wrote:

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

Posted from my mobile device

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

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Joined: 28 Apr 2014

Posts: 141

Re: A perfect square is defined as the square of an integer and [#permalink]

21 Jul 2014, 06:12

Mackieman wrote:

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)

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

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Bunuel

Math Expert

Joined: 02 Sep 2009

Posts: 92747

Re: A perfect square is defined as the square of an integer and [#permalink]

21 Jul 2014, 06:33

Expert Reply

himanshujovi wrote:

Mackieman wrote:

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)

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

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

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Joined: 09 Jul 2015

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Concentration: Finance, Technology

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Re: A perfect square is defined as the square of an integer and [#permalink]

27 Jul 2015, 03:56

1

Kudos

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

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

P

JeffTargetTestPrep

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Posts: 3043

Re: A perfect square is defined as the square of an integer and [#permalink]

18 Aug 2017, 09:07

Expert Reply

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

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
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PranayBudime

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Re: A perfect square is defined as the square of an integer and [#permalink]

19 Aug 2017, 02:02

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

B

andr3

Intern

Joined: 22 Oct 2017

Posts: 4

Re: A perfect square is defined as the square of an integer and [#permalink]

09 Jun 2018, 03:02

Bunuel wrote:

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.

Answer: B.

amod243 wrote:

hi Mackieman

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

because 0 also satisfy the condition..

n cannot be 0 as given that n is a positive integer.

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

3176401

Intern

Joined: 02 Oct 2018

Posts: 1

Re: A perfect square is defined as the square of an integer and [#permalink]

17 Jul 2019, 22:53

Where is this question taken from?

B

prashu1701

Intern

Joined: 02 Mar 2019

Posts: 5

Re: A perfect square is defined as the square of an integer and [#permalink]

18 Jul 2019, 00:19

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

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Re: A perfect square is defined as the square of an integer and [#permalink]

21 Mar 2022, 03:09

Expert Reply

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

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

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hellypatel

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Joined: 12 May 2023

Posts: 8

Re: A perfect square is defined as the square of an integer and [#permalink]

01 Jun 2023, 09:24

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

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

so B

gmatclubot

Re: A perfect square is defined as the square of an integer and [#permalink]

01 Jun 2023, 09:24

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