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

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A perfect square is defined as the square of an integer and [#permalink] New post 22 Jan 2011, 03:49
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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
[Reveal] Spoiler: OA

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Re: A perfect square is defined as the square of an integer and [#permalink] New post 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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Re: A perfect square is defined as the square of an integer and [#permalink] New post 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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Re: A perfect square is defined as the square of an integer and [#permalink] New post 22 Jan 2011, 08:12
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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 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] New post 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.
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Re: A perfect square is defined as the square of an integer and [#permalink] New post 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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Re: A perfect square is defined as the square of an integer and [#permalink] New post 18 Feb 2011, 13:59
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Re: A perfect square is defined as the square of an integer and [#permalink] New post 18 Jul 2014, 23:30
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Re: A perfect square is defined as the square of an integer and [#permalink] New post 19 Jul 2014, 16:21
Bunuel - how did you come up with x^6?

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A perfect square is defined as the square of an integer and [#permalink] New post 19 Jul 2014, 22:52
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bankerboy30 wrote:
Bunuel - how did you come up with x^6?

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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
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Re: A perfect square is defined as the square of an integer and [#permalink] New post 21 Jul 2014, 05:12
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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Re: A perfect square is defined as the square of an integer and [#permalink] New post 21 Jul 2014, 05:33
Expert's post
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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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: A perfect square is defined as the square of an integer and   [#permalink] 21 Jul 2014, 05:33
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