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Is C square of an Integer? 1) Sum of the factors of integer C is ODD

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GMATinsight
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Is C square of an Integer? 1) Sum of the factors of integer C is ODD [#permalink] New post   27 Apr 2018, 01:46
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Is C square of an Integer?

1) Sum of the factors of integer C is ODD
2) Number of distinct factors of Integer C is ODD

Source: www.GMATinsight.com
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Re: Is C square of an Integer? 1) Sum of the factors of integer C is ODD [#permalink] New post   27 Apr 2018, 02:06
GMATinsight wrote:
Is C square of an Integer?

1) Sum of the factors of integer C is ODD
2) Number of distinct factors of Integer C is ODD

Source: https://www.GMATinsight.com


St1 : Sum of the factors of integer C is ODD

Factors of 9 = 1, 3, 9; so sum : ODD
Factors of 8 = 1, 2, 4, 8 Sum : ODD

Hence statement 1 insuff

St2 : Number of distinct factors of Integer C is ODD
For a int N = \(x^ay^b..\) the number of factors are \((a+1)*(b+1)...\)
So if \((a+1)*(b+1)...\) is odd, this means each \((a+1), (b+1),...\) is odd

or A, B,... are even
Hence the number \(N = x^{even}*Y^{even}...\)

So We can say Number is square of number.

Hence Option B
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Re: Is C square of an Integer? 1) Sum of the factors of integer C is ODD [#permalink] New post   27 Apr 2018, 03:34
GMATinsight wrote:
Is C square of an Integer?

1) Sum of the factors of integer C is ODD
2) Number of distinct factors of Integer C is ODD

Source: https://www.GMATinsight.com


Let us try to solve this by trying numbers :

let n = 4. we know n is a square of integer 2.

1) Sum of the factors of integer C is ODD. Not sufficient 1 + 2 + 4 = 7 ( true for 4) 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91 ( true for 36) 1 + 17 + 289 ( true for 289 ).
Although it is true for perfect squares, it is also true for other which are not perfect squares

ex. 1 + 2 + 4 + 8 = 15 ( also true for 8)

2) Number of distinct factors of Integer C is ODD Suff this is a property which is easily seen to be true. all the factors will pair out and the perfect square will remain. look at the factors of 36 to verify this

Hence Option (B) is our answer.

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Re: Is C square of an Integer? 1) Sum of the factors of integer C is ODD [#permalink] New post   22 Aug 2018, 17:55
statement 1:
we can only infer that no. of odd prime factors is odd but we don't know the no. of even prime factor (2):
eg. 3^2*2^3 has 3 odd factors and 9 even factors but the sum of the factors is odd
3^2*2^4 has 3 odd factors and 12 even factors but the sum of the factors is still odd
statement 2:
assume C=A^a*B^b*E^e
no. of factors: (a+1)(b+1)(c+1)
a,b,e all have to be even no. to have odd no. of distince factors. hence C is a squre
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Re: Is C square of an Integer? 1) Sum of the factors of integer C is ODD [#permalink]
22 Aug 2018, 17:55
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