NOTE: I have added brackets to the question to avoid ambiguity.
M7MBA wrote: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
