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

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SVP
Joined: 16 Jul 2009
Posts: 1508

Kudos [?]: 1401 [0], given: 2

Schools: CBS
WE 1: 4 years (Consulting)
Is the positive integer N a perfect square? (1) The number [#permalink]

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27 Jul 2009, 15:27
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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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Kudos [?]: 1401 [0], given: 2

Senior Manager
Joined: 17 Jul 2009
Posts: 286

Kudos [?]: 44 [0], given: 9

Concentration: Nonprofit, Strategy
GPA: 3.42
WE: Engineering (Computer Hardware)

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27 Jul 2009, 15:36
say number is 6,

2+3+6+1 = even

number of distinct factors = 4 = even....6 is not a perfect square

Kudos [?]: 44 [0], given: 9

Manager
Joined: 03 Jul 2009
Posts: 106

Kudos [?]: 93 [1], given: 13

Location: Brazil

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27 Jul 2009, 16:58
1
KUDOS
D

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

D

PS.: If you liked the explanation, consider a kudo. I want to access the GMATClub tests!!!

Kudos [?]: 93 [1], given: 13

Re: perfect square   [#permalink] 27 Jul 2009, 16:58
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