Q.)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.
Someone explain..
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For most takers, the best approach would be to plug in values.akshatgupta87 wrote:Q.)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.
Someone explain..
Since the statements deal with even versus odd, let's list a few even and a few odd perfect squares:
1, 4, 9, 16, 81.
Statement 1: The number of distinct factors of N is even.
Factors of 1 = 1.
Factors of 4 = 1,2,4 = 3.
Factors of 9 = 1,3,9 = 3.
Factors of 16 = 1,2,4,8,16 = 5.
Factors of 81 = 1,3,9,27,81 = 5.
The pattern indicates that a perfect square has an odd number of distinct factors.
Thus N is not a perfect square.
Sufficient.
Statement 2: The sum of all distinct factors of N is even.
Sum of the factors of 1 = 1.
Sum of the factors of 4 = 1+2+4 = 7.
Sum of the factors of 9 - 1+3+9 = 13.
Sum of the factors of 16 = 1+2+4+8+16 = 31.
Sum of the factors of 81 = 1+3+9+27+81 = 121.
The pattern indicates that the sum of the factors of a perfect square is odd.
Thus N is not a perfect square.
Sufficient.
The correct answer is D.
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It can be useful to understand when a number will have an even number of divisors, and when a number will have an odd number of divisors. If you list all of the factors of, say, 12, they are all in pairs which give a product of 12:akshatgupta87 wrote:Q.)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.
Someone explain..
1*12
2*6
3*4
So most numbers have an even number of factors. Perfect squares are an exception, however; they will always have one factor which is not in a 'pair'. Taking 36 for example:
1*36
2*18
3*12
4*9
6
We have an odd number of factors, because 6 is not in a pair.
The upshot is:
- perfect squares always have an odd number of factors
- if a number is not a perfect square, it has an even number of factors
If you know this, then you can see instantly that Statement 1 is sufficient here.
Statement 2 is also sufficient, but it tests something you'd never be expected to know for the GMAT. Since almost every test taker would find it difficult to establish that Statement 2 is sufficient within two minutes without resorting to number-picking, this is clearly a prep company question, and not a real GMAT question. I've explained why Statement 2 is sufficient in other posts, so you can probably find a solution with a forum search (this question has been posted here quite a few times) - you just need to establish that a perfect square has an odd number of odd divisors - but understanding why Statement 2 is sufficient is not going to help you on your GMAT, so I won't go into further detail here.
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