sumanr84 wrote: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.
All the numbers can be expressed in a way
a^x*b^y*c^z....
a,b,c being prime and x,y,z ...an integer greater than or equal to 1
If the number is a square and x,y,z... all will be even numbers
so, the number of distinct factors will be (x+1)*(y+1)*(z+1).... Will be odd
If the number is not a square one of x,y,z... will be even thus making the num of factors even (x+1)*(y+1)*(z+1)....
So
1 is sufficient to conclude that the number is not a square
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as far as 2 goes it says that sum of all distinct factors = even
For all squares sum of the factors = odd
that is because
All the squares can be expressed in a way
a^x*b^y*c^z....
a,b,c being prime and x,y,z ...all are even numbers
now sum of factors= (x^0+x^1+.....x^a) (y^0+y^1+....y^b)......
Now (x^0+x^1+.....x^a) is odd because it is 'a' is even
same goes for (y^0+y^1+....y^b)
hence sum of factors for a perfect square is always odd
So 2 is also sufficient to conclude that the number is not a square
[I know the explanation contains a lot of math, I am extremely sorry and this will most probably not be a GMAT Question]
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