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.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Official problems about factors are generally constrained to POSITIVE factors.
If the problem posted above were to appear on the GMAT, it would probably read as follows:

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, test 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, 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.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


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.

From the original condition we have 1 variable (n) and we need 1 equation to match the number of variables and equations. Since we have 1 each in 1) and 2), D is likely the answer.

In actual calculation,
In case of 1), the number of factors = number of indices + 1 and since even number = odd + 1, n = prime^odd thus the answer is no. Therefore it is sufficient.
In case of 2), for the addition of all factors to be an even number we need primes^odd (1+p+p^2+....+p^odd=even, odd+odd=even). Thus the answer is no. Therefore it is sufficient.
The answer is D.


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Max@Math Revolution wrote:Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


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.

From the original condition we have 1 variable (n) and we need 1 equation to match the number of variables and equations. Since we have 1 each in 1) and 2), D is likely the answer.

In actual calculation,
In case of 1), the number of factors = number of indices + 1 and since even number = odd + 1, n = prime^odd thus the answer is no. Therefore it is sufficient.
In case of 2), for the addition of all factors to be an even number we need primes^odd (1+p+p^2+....+p^odd=even, odd+odd=even). Thus the answer is no. Therefore it is sufficient.
The answer is D.


If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
www.mathrevolution.com
l The one-and-only World's First Variable Approach for DS and IVY Approach for PS that allow anyone to easily solve GMAT math questions.

l The easy-to-use solutions. Math skills are totally irrelevant. Forget conventional ways of solving math questions.

l The most effective time management for GMAT math to date allowing you to solve 37 questions with 10 minutes to spare

l Hitting a score of 45 is very easy and points and 49-51 is also doable.

l Unlimited Access to over 120 free video lessons at https://www.mathrevolution.com/gmat/lesson

l Our advertising video at https://www.youtube.com/watch?v=R_Fki3_2vO8

Can you explain in more depth
"From the original condition we have 1 variable (n) and we need 1 equation to match the number of variables and equations. Since we have 1 each in 1) and 2), D is likely the answer."

I dont understand

oquiella wrote: Can you explain in more depth
"From the original condition we have 1 variable (n) and we need 1 equation to match the number of variables and equations. Since we have 1 each in 1) and 2), D is likely the answer."

I dont understand

Neither do I. This logic just isn't correct. For instance, suppose that I asked

"Is integer N an even prime?" and gave you two statements: S1:: N > 2 and S2:: N > 1. By the logic given above, the answer is likely D ... but of course it isn't.

Simply matching the number of variables to the number of equations is not a safe strategy on most GMAT problems.