dddanny2006 wrote: If d is a positive integer, is sqrt(d) an integer?
(1) sqrt(9d) is an integer.
(2) sqrt(10d) is not an integer.

Asde: This is very similar to this question: https://www.beatthegmat.com/og-quant-rev ... tml#688190


Target question: Is sqrt(d) an integer?

Given: d is a positive integer

Statement 1: sqrt(9d) is an integer
IMPORTANT CONCEPT: If K is an integer, then sqrt(K) will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
sqrt(144) = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
sqrt(1600) = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
sqrt(441) = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
sqrt(12) = some non-integer, and 12 = (2)(2)(3)[two 2's and one 3]

So, if sqrt(9d) is an integer, then the prime factorization of 9d has an even number of each prime.
Since 9d = (3)(3)(d) we can see that the prime factorization of d must have an even number of each prime.
If the prime factorization of d has an even number of each prime, then sqrt(d) must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: sqrt(10d) is NOT an integer.
There are several values of d that meet this condition. Here are two:
Case a: d = 4. This means that sqrt(10d) = sqrt(40), which is not an integer. In this case, sqrt(d) is an integer.
Case b: d = 5. This means that sqrt(10d) = sqrt(50), which is not an integer. In this case, sqrt(d) is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

dddanny2006 wrote:If d is a positive integer,is sqrt of d an integer?

1.Sqrt of 9d is an integer
2.Sqrt of 10d is NOT an integer.


Now this is how I understand it for Statement 1.

Let the integer be 2,that implies 9d=4 that implies d=4/9 Sqrt of 4/9 is not an integer--------NO

Let the integer be 9,that implies 9d=81,that implies d=9 Sqrt of 9 is 3 which is a integer------YES


Therefore it is insufficient.

Same is the case with Statement 2 as well and hence I think the answer should be E

The book says the answer is A.

Why is my approach incorrect??

Please clarify,


Dan

Hey Dan,

In your solution, you let d = 4/9 , but the question states that d is a positive integer.

Cheers,
Brent

Hi dddanny2006,

In this DS question, we are told that d is a positive integer and we're asked if root(d) is an integer? This is a YES/NO question.

Fact 1: root(9d) is an integer.

Using square root rules, we can simplify this expression by pulling out the 9, which gives us...

3(root(d)) = an integer

Since we're restricted to positive integers, d can ONLY be a perfect square = 1, 4, 9, 16, etc.
If d = 1 then the answer to the question is YES
If d = 4 then the answer to the question is YES
If d = 9 then the answer to the question is YES
etc.
Consistent = SUFFICIENT

To beat DS questions, you MUST pay attention to what you're told.

GMAT assassins aren't born, they're made,
Rich