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Is the positive integer N a perfect square? (1) The number

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Joined: 16 Jul 2009
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Is the positive integer N a perfect square? (1) The number [#permalink] New post 27 Jul 2009, 14:27
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Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even
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Senior Manager
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Joined: 17 Jul 2009
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Concentration: Nonprofit, Strategy
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Re: perfect square [#permalink] New post 27 Jul 2009, 14:36
say number is 6,

2+3+6+1 = even

number of distinct factors = 4 = even....6 is not a perfect square
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Re: perfect square [#permalink] New post 27 Jul 2009, 15:58
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After checking some perfect squares I found something very interesting:

N, a perfect square, can be 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 ...

1) If you pick any of this numbers you will see that "The number of distinct factors of N is always odd".
16 - 1, 2, 4, 8, 16
81 - 1, 3, 9, 27, 81
64 - 1, 2, 4, 8, 16, 32, 64

So, as all perfect squares have an odd number of distinct factors, we know that N is not a perfect square
SUFFIC

2) Again, very interesting.
In all the perfect squares, the sum of its factors is always odd. So again we know that N is not a perfect square.
SUFFIC

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PS.: If you liked the explanation, consider a kudo. ;) I want to access the GMATClub tests!!!
Re: perfect square   [#permalink] 27 Jul 2009, 15:58
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Is the positive integer N a perfect square? (1) The number

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