if d is a positive integer, is √d an integer ?

[M7MBA]

if d is a positive integer, is √d an integer?

(1) √9d is an integer

(b) √10d is not an integer.

The OA is the option A.

Why is sufficient the statement (1)? And why the second one is not sufficient? I am confused.

[Brent@GMATPrepNow]

NOTE: I have added brackets to the question to avoid ambiguity.

if d is a positive integer, is √d an integer?

(1) √(9d) is an integer
(b) √(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 √d an integer?

Given: d is a positive integer

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

So, if √(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 √d MUST be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: √(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 √(10d) = √(40), which is not an integer. In this case, √d is an integer.
Case b: d = 5. This means that √(10d) = √(50), which is not an integer. In this case, √d is NOT an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

[Jeff@TargetTestPrep]

if d is a positive integer, is √d an integer?

(1) √9d is an integer

(b) √10d is not an integer.

We are given that d is a positive integer and need to determine if √d is an integer.

Statement One Alone:

√9d is an integer

Simplifying we have:

√9 x √d = integer

3 x √d = integer

The only way for 3 x √d to be an integer, is for √d to be an integer. Statement one alone is sufficient to answer the question.

Statement Two Alone:

√10d is not an integer.

Simplifying we have:

√10 x √d is not an integer.

We see that if d = 4, then √d is an integer, but if d =5, then √d is not.

Answer: A