rishianand7 wrote:Is the square root of the positive integer X an integer?
(1) The sum of the distinct factors of X is odd.
(2) X has an odd number of distinct 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 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 factors.
melguy mentions a useful property: "If X has an odd number of distinct factors, then X is the square of an integer" (
NOTE: we'll examine this property at the end of the post).
Given this 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
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Now, let's take a closer look at melguy's useful property:
"If X has an odd number of distinct factors, then X is the square of an integer"
Why is this?
Notice that, when we find factors, we can typically do so by finding pairs of values that have a some particular product.
So, for example, to find the factors of 12, we can observe that (1)(12)=12, (2)(6)=12, and (3)(4)=12. So, the factors of 12 are 1,2,3,4,6 and 12.
So, if we have several pairs of factors (like we have above), then we will get an EVEN number of factors.
Under what circumstances will we get an ODD number of factors?
This will occur when one pair of factors has IDENTICAL VALUES.
For example, if we use pairs of values to find the factors of 36, we get (1)(36)=36, (2)(18)=36, (3)(12)=36, (4)(9)=36, (
6)(
6)=36.
So, we have 4 pairs of DIFFERENT values, and 1 pair of IDENTICAL values. This gives us an ODD number of factors.
So, the factors of 36 are: 1, 2, 3, 4,
6, 9, 12, 18, 36
Cheers,
Brent
