Let's solve this GMAT-style question
Is the square root of the positive integer X an integer?
(1) The sum of the distinct positive factors of X is odd.
(2) X has an odd number of distinct positive factors.
Target question: Is the square root of the positive integer X an integer?
This question is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100
If the square root of X is an integer, what
must be true about X?
X
must be the square of an integer (e.g., 1, 4, 9, 16, etc.)
Rephrased target question: Is X the square of an integer?
Statement 1: The sum of the distinct positive factors of X is odd.
There are many values that meet this condition. Here are two:
Case a: X = 1 (1 has only 1 as its factor, so the sum = 1, which is odd). In this case
X is the square of an integer
Case b: X = 2 (the factors of 2 are 1 and 2, so the sum = 3, which is odd). In this case
X is not the square of an integer
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: X has an odd number of distinct positive factors.
There's a nice rule that says,
"If positive integer k has an odd number of distinct factors, then k is the square of an integer"
Some examples: The factors of 9 are {1,3,9}. There is an
odd number of factors and 9 is a
square
The factors of 36 are {1,2,3,4,6,9,12,18,36}. There is an
odd number of factors and 36 is a
square
The factor of 1 is {1} There is an
odd number of factors and 1 is a
square
Conversely, the factors of 6 are {1,2,3,6}. There is an
even number of factors and 6 is
not a square
The factors of 10 are {1,2,5,10}. There is an
even number of factors and 10 is
not a square
The factors of 20 are {1,2,4,5,10,20}. There is an
even number of factors and 20 is
not a square
So, given the
above property, we can be certain that
X is the square of an integer
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
B
Cheers,
Brent
