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# Which of the following is not a perfect square

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Manager
Joined: 19 Aug 2009
Posts: 77
Which of the following is not a perfect square  [#permalink]

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03 Nov 2009, 23:22
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95% (hard)

Question Stats:

24% (01:43) correct 76% (01:17) wrong based on 82 sessions

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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
Senior Manager
Joined: 31 Aug 2009
Posts: 331
Location: Sydney, Australia
Re: Which of the following is not a perfect square  [#permalink]

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03 Nov 2009, 23:44
1
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 
Manager
Joined: 19 Aug 2009
Posts: 77
Re: Which of the following is not a perfect square  [#permalink]

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03 Nov 2009, 23:52
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 
Senior Manager
Joined: 31 Aug 2009
Posts: 331
Location: Sydney, Australia
Re: Which of the following is not a perfect square  [#permalink]

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04 Nov 2009, 00:31
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.
Tuck School Moderator
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Posts: 224
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)
Re: Which of the following is not a perfect square  [#permalink]

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04 Nov 2009, 00:32
Quote:
A perfect square will have exactly an even number of each of its factors.

A simple point, but worth remembering, Kudos!
Math Expert
Joined: 02 Sep 2009
Posts: 64147
Re: Which of the following is not a perfect square  [#permalink]

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04 Nov 2009, 22:57
1
1
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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Tuck School Moderator
Joined: 20 Aug 2009
Posts: 224
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)
Re: Which of the following is not a perfect square  [#permalink]

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04 Nov 2009, 23:07
Quote:
A perfect square will have exactly an even number of each of its factors.

yeah, it holds only for prime factors
Tuck School Moderator
Joined: 20 Aug 2009
Posts: 224
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)
Re: Which of the following is not a perfect square  [#permalink]

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04 Nov 2009, 23:15
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 . . .
Senior Manager
Joined: 31 Aug 2009
Posts: 331
Location: Sydney, Australia
Re: Which of the following is not a perfect square  [#permalink]

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05 Nov 2009, 21:02
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
Director
Joined: 23 Apr 2010
Posts: 501
Re: Which of the following is not a perfect square  [#permalink]

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02 Jan 2011, 07:38
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.
Math Expert
Joined: 02 Sep 2009
Posts: 64147
Re: Which of the following is not a perfect square  [#permalink]

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02 Jan 2011, 07:47
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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Director
Joined: 23 Apr 2010
Posts: 501
Re: Which of the following is not a perfect square  [#permalink]

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02 Jan 2011, 07:55
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?
Math Expert
Joined: 02 Sep 2009
Posts: 64147
Re: Which of the following is not a perfect square  [#permalink]

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02 Jan 2011, 08:04
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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Director
Joined: 23 Apr 2010
Posts: 501
Re: Which of the following is not a perfect square  [#permalink]

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02 Jan 2011, 08:07
OK. Thanks. I guess I have to study it.
Intern
Joined: 24 Mar 2017
Posts: 1
Re: Which of the following is not a perfect square  [#permalink]

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09 Aug 2017, 11:21
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?

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

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26 Jun 2018, 17:18
virtualanimosity wrote:
Which of the following is not a perfect square

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

We know that a perfect square must end with 0, 1, 4, 5, 6, or 9, so we can see choice 2 is not a perfect square.

Now let’s check choice 1. Since 310^2 = 96,100 and 320^2 = 102,400, then, if 100,856 is a perfect square, it must be the square of either 314 or 316 since either one of these integers will produce a units digit of 6 when squared. However, since 314^2 = 98,596 and 316^2 = 99,856, we see that choice 1 is a not a perfect square, either.

Since both choices 1 and 2 are not perfect squares, we will conclude that the correct answer must be choice 4 (even though we have not verified that choice 3 is not a perfect square).

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Re: Which of the following is not a perfect square   [#permalink] 26 Jun 2018, 17:18