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Which of the following is not a perfect square

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Which of the following is not a perfect square [#permalink]

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Which of the following is not a perfect square

1. 100856
2. 325137
3. 945729
4. All of these

Hint- value of perfect sq has to end in 1,4,5,6,9...so option 2 is definately not a perfect square

But how to find the rest...is ther any shortcut or will we have to find the prime factors of all n bring them to standard form
[Reveal] Spoiler: OA

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Re: Which of the following is not a perfect square [#permalink]

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New post 04 Nov 2009, 00:44
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A perfect square will have exactly an even number of each of its factors.
81 for example:
81 = 9x9 = 3x3x3x3 which is essentially 2 sets of 3x3
Or:
36 = 6x6 = 2x3x2x3 which is 2 sets of 2x3

1. 100856
2. 325137
3. 945729
4. All of these

You mention option 2 is not correct, so neither is option 4.
Statement 1: 100856
You can only factor it by 2 three times. This is not an even amount of 2s so it cannot be a perfect square.
Statement 3: 945729
You can factor 3 three times. This is not an even amount of 3s so it cannot be a perfect square

So if I’m not wrong then none of these are perfect square? Which seems odd as that’s not an answer option. Or I could be wrong 

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Re: Which of the following is not a perfect square [#permalink]

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New post 04 Nov 2009, 00:52
Ans is
optn 4 as it says 'all of these' meaning all of these r not perfect squares


yangsta8 wrote:
A perfect square will have exactly an even number of each of its factors.
81 for example:
81 = 9x9 = 3x3x3x3 which is essentially 2 sets of 3x3
Or:
36 = 6x6 = 2x3x2x3 which is 2 sets of 2x3

1. 100856
2. 325137
3. 945729
4. All of these

You mention option 2 is not correct, so neither is option 4.
Statement 1: 100856
You can only factor it by 2 three times. This is not an even amount of 2s so it cannot be a perfect square.
Statement 3: 945729
You can factor 3 three times. This is not an even amount of 3s so it cannot be a perfect square

So if I’m not wrong then none of these are perfect square? Which seems odd as that’s not an answer option. Or I could be wrong 

Kudos [?]: 270 [0], given: 46

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Re: Which of the following is not a perfect square [#permalink]

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New post 04 Nov 2009, 01:31
Ahh yes you are right I didn't read it properly. Also to answer your original question I guess there is no need to factor all prime factors, just factor a few until you find that there are not an even number.

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Re: Which of the following is not a perfect square [#permalink]

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New post 04 Nov 2009, 01:32
Quote:
A perfect square will have exactly an even number of each of its factors.


A simple point, but worth remembering, Kudos!

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Re: Which of the following is not a perfect square [#permalink]

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New post 04 Nov 2009, 23:57
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yangsta8 wrote:
A perfect square will have exactly an even number of each of its factors.
81 for example:
81 = 9x9 = 3x3x3x3 which is essentially 2 sets of 3x3
Or:
36 = 6x6 = 2x3x2x3 which is 2 sets of 2x3


I think the red part needs some clarification:

A perfect square will have exactly an even number of each of its factors.

That's not true.

I think you meant that perfect square n^2, has even number of prime factors of n or in other words prime factors of n have even powers.

In your example: 81=3*3*3*3=3^4, but 81 has the factor 27, what about it? There are three 27 in 81.
Or 36=6*6=2^2*3^2 but what about 12 which is factor of 36? There are three 12 in 36.

There are some tips about the perfect square though:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
4. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
5. Perfect square always has even number of powers of prime factors.

Hope it helps.
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Re: Which of the following is not a perfect square [#permalink]

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New post 05 Nov 2009, 00:07
Quote:
A perfect square will have exactly an even number of each of its factors.


yeah, it holds only for prime factors

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Re: Which of the following is not a perfect square [#permalink]

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New post 05 Nov 2009, 00:15
Man, there should be a way to add some posts to favorites, to "memories" or something like that. The only method available is to save permalinks . . .

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Re: Which of the following is not a perfect square [#permalink]

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New post 05 Nov 2009, 22:02
shalva wrote:
Quote:
A perfect square will have exactly an even number of each of its factors.


yeah, it holds only for prime factors


Yes you are right. Thanks for the clarification :)

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Re: Which of the following is not a perfect square [#permalink]

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New post 02 Jan 2011, 08:38
Bunuel, is the concept of a perfect square tested on the GMAT? I have come across it several times on this forum, but I am not sure if it is tested on the GMAT. Thank you.

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Re: Which of the following is not a perfect square [#permalink]

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New post 02 Jan 2011, 08:47
nonameee wrote:
Bunuel, is the concept of a perfect square tested on the GMAT? I have come across it several times on this forum, but I am not sure if it is tested on the GMAT. Thank you.


What do you mean by "the concept of a perfect square"??? A perfect square, is just an integer that can be written as the square of some other integer, for example 16=4^2, is a perfect square.
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Collection of Questions:
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Re: Which of the following is not a perfect square [#permalink]

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New post 02 Jan 2011, 08:55
What I mean is that I've come across several questions here on the forum that have a phrase "perfect square", and I have never heard of it before. So the reason I'm asking is if I have to pay attention to these questions (and read about some useful properties of perfect squares) or not?

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Re: Which of the following is not a perfect square [#permalink]

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New post 02 Jan 2011, 09:04
nonameee wrote:
What I mean is that I've come across several questions here on the forum that have a phrase "perfect square", and I have never heard of it before. So the reason I'm asking is if I have to pay attention to these questions (and read about some useful properties of perfect squares) or not?


Again: a perfect square, is just an integer that can be written as the square of some other integer, for example 16=4^2, is a perfect square. So perfect square is just a name of such integers as 1, 4, 9, 16, ... There are several properties of a perfect square, which might be useful while solving specific GMAT questions. Now, it's up to you to decide whether to study these question and properties or not.
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Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

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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.


What are GMAT Club Tests?
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Re: Which of the following is not a perfect square [#permalink]

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New post 02 Jan 2011, 09:07
OK. Thanks. I guess I have to study it.

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Re: Which of the following is not a perfect square [#permalink]

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New post 09 Aug 2017, 12:21
Bunuel wrote:
yangsta8 wrote:
A perfect square will have exactly an even number of each of its factors.
81 for example:
81 = 9x9 = 3x3x3x3 which is essentially 2 sets of 3x3
Or:
36 = 6x6 = 2x3x2x3 which is 2 sets of 2x3


I think the red part needs some clarification:

A perfect square will have exactly an even number of each of its factors.

That's not true.

I think you meant that perfect square n^2, has even number of prime factors of n or in other words prime factors of n have even powers.

In your example: 81=3*3*3*3=3^4, but 81 has the factor 27, what about it? There are three 27 in 81.
Or 36=6*6=2^2*3^2 but what about 12 which is factor of 36? There are three 12 in 36.

There are some tips about the perfect square though:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
4. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
5. Perfect square always has even number of powers of prime factors.

Hope it helps.


Could you please prove these rules?

I know only that the total number of factors (of a perfect square) is odd. Could you clarify the others?

Thanks in advance.

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Re: Which of the following is not a perfect square   [#permalink] 09 Aug 2017, 12:21
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