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.
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As Brent mentions in his post below, the problem should read as follows:
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, to satisfy statement 1, N must not be 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, to satisfy statement 2, N must not be a perfect square.
Sufficient.
The correct answer is D.
Since the statements deal with even versus odd, test a few even and a few odd perfect squares: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.
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, to satisfy statement 1, N must not be 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, to satisfy statement 2, N must not be a perfect square.
Sufficient.
The correct answer is D.
Last edited by GMATGuruNY on Sun Jan 04, 2015 10:19 am, edited 1 time in total.
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Are you sure you correctly transcribed the question?akhilsuhag 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.
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:
So, for example, the -2 is a factor of 6 since 6 = (-2)(-3)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.
No 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 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
Last edited by Brent@GMATPrepNow on Mon Jan 05, 2015 11:28 am, edited 1 time in total.
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One little hiccup in Brent's otherwise great explanation: that should be 6 = (-2)(-3). (In case anyone was confused ... it's an abstract topic already!)Brent@GMATPrepNow wrote:So, for example, the -2 is a factor of 6 since 6 = (-2)(3)
I am surprised the notion of negative factors would come up on the GMAT; I had always assumed that the testwriters treated factors as implicitly positive. (Is there any official question that turns on negative factors?)
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Good catch, Matt. Fixed now.
I don't think the testwriters treat factors as implicitly positive.
From what I've seen, there's always some text that restricts the conversation to positive values only.
For example (from the Official Guide),
For example (from the Official Guide),
Cheers,
Brent
I don't think the testwriters treat factors as implicitly positive.
From what I've seen, there's always some text that restricts the conversation to positive values only.
For example (from the Official Guide),
The only time they don't make this restriction is when the correct answer is unaffected by negative factors/divisors.If n=4p where p is a prime number greater than 2, how many different positive even divisors does n have including n
A.2
B.3
C.4
D.6
E.8
For example (from the Official Guide),
Here, we're talking about factors that are prime, and prime numbers are defined as being positive only.How many prime numbers between 1 and 100 are factors of 7,150 ?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
Cheers,
Brent