Is the positive integer N a perfect square?
(1) The number of distinct positive factors of N is even.
(2) The sum of all distinct positive factors of N is even.
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 results above indicate that a perfect square has an ODD number of distinct factors.
Since N has an EVEN number of distinct factors, 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 results above indicate that the sum of the factors of a perfect square is ODD.
Since the sum of the distinct factors of N is EVEN, N is NOT a perfect square.
SUFFICIENT.
The correct answer is
D.
Note:
A perfect square has an ODD number of distinct positive factors.
An integer that is NOT a perfect square has an EVEN number of distinct positive factors.
A test-taker who knows these two properties can determine that Statement 1 is sufficient without doing any work.
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