Data Sufficiency

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.

As it is currently worded, the answer to this question is E.
The answer WOULD be D, if we changed "distinct factors" to "POSITIVE distinct factors."

When asking questions about factors (aka divisors), the GMAT typically restricts the discussion to POSITIVE factors/divisors. If we don't specify such a restriction, then we must also consider negative factors.

From the Official Guide:

An integer is any number in the set {. . . -3, -2, -1, 0, 1, 2, 3, . . .}.
If x and y are integers and x ≠ 0, then x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible by x or to be a multiple of x.
For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a divisor of 28 since there is no integer n such that 28 = 8n.

So, for example, the -2 is a factor of 6 since 6 = (-2)(-3)

No onto the question....

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Target question: Is the positive integer N a perfect square?

Statement 2: The number of distinct factors of N is even
There are infinitely many values of N that satisfy this condition. Here are two:
Case a: N = 3. The distinct factors of N are {-3, -1, 1, 3}. As you can see, there is an even number of distinct factors of N. In this case N is NOT a perfect square
Case b: N = 4. The distinct factors of N are {-4, -2, -1, 1, 2, 4}. As you can see, there is an even number of distinct factors of N. In this case N IS perfect square
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The sum of all distinct factors of N is even
There are infinitely many values of N that satisfy this condition. Here are two:
Case a: N = 3. The distinct factors of N are {-3, -1, 1, 3}, so the sum = (-3) + (-1) + 1 + 3 = 0. The sum of the distinct factors = 0, which is EVEN. In this case N is NOT a perfect square
Case b: N = 4. The distinct factors of N are {-4, -2, -1, 1, 2, 4}, so the sum = (-4) + (-2) + (-1) + 1 + 2 + 4 = 0. The sum of the distinct factors = 0, which is EVEN. In this case N IS a perfect square
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
In both cases, I showed that N COULD equal 3 or 4.
So, when we combine the statements, N COULD still equal 3 or 4.
3 is NOT a perfect square, and 4 IS a perfect square.
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

Matt@VeritasPrep wrote:

Brent@GMATPrepNow wrote:If we don't specify such a restriction, then we must also consider negative factors.

What official problems turn on negative factors?

Good point. I don't know of any such questions.

Likewise, I've never seen an official question that didn't restrict factors to positive factors. Have you seen any?

The bigger question might be, "Would the test-makers ever create a question that requires the test-taker to consider the possibility that a factor can be negative?" Unless we can decisively answer "no," then I think my point has some merit. That said, I'm happy to be convinced otherwise

Cheers,
Brent