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

Question

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

RohitGoyal1123 · Answer

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

Gladiator59 · Answer

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.

Best,
Gladi

shmba · Answer

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

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

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

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