oa7 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.
Are you sure you correctly transcribed the question?
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:
Quote:
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)
Now onto the question....
----------------------------------------
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 squareCase 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 squareSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The sum of all distinct factors of N is evenThere 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 squareCase 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 squareSince 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
_________________
Brent Hanneson – Creator of gmatprepnow.com
I’ve spent the last 20 years helping students overcome their difficulties with GMAT math, and the biggest thing I’ve learned is…
Many students fail to maximize their quant score NOT because they lack the skills to solve certain questions but because they don’t understand what the GMAT is truly testing -
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