Question

\( x \) is a positive integer. Is the GCD of 150 and \( x \) a prime number? Statement (I): \( x \) is a prime number.
Statement (II): \( x < 4 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

In Data Sufficiency problems, do not try to determine the exact value of the variable unless necessary. Instead, check whether the information provided guarantees a unique YES or NO answer. If every possible value satisfying the statements leads to the same conclusion, the statements are sufficient.

Answer 1

The Correct Option is C

Concept: The Greatest Common Divisor (GCD) of two numbers is the largest positive integer that divides both numbers exactly. A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself. The prime factorization of 150 is: \[ 150 = 2 \times 3 \times 5^2 \] To determine whether the GCD of 150 and \(x\) is prime, we must know enough information about \(x\) so that the answer is definitely "Yes" or definitely "No".

Step 1: Examine Statement (I) alone. Statement (I) says: \[ x \text{ is a prime number} \] Since \(x\) can be any prime number, several possibilities arise: \[ x=2 \] Then, \[ \gcd(150,2)=2 \] and 2 is a prime number. However, if \[ x=7 \] then \[ \gcd(150,7)=1 \] and 1 is not a prime number. Thus, Statement (I) can produce both a prime GCD and a non-prime GCD. Therefore, Statement (I) alone is not sufficient.

Step 2: Examine Statement (II) alone. Statement (II) says: \[ x<4 \] Since \(x\) is a positive integer, \[ x\in\{1,2,3\} \] Check each case: For \(x=1\), \[ \gcd(150,1)=1 \] which is not prime. For \(x=2\), \[ \gcd(150,2)=2 \] which is prime. For \(x=3\), \[ \gcd(150,3)=3 \] which is also prime. Since different values of \(x\) lead to different answers, Statement (II) alone is not sufficient.

Step 3: Combine Statements (I) and (II). From Statement (I): \[ x \text{ is prime} \] From Statement (II): \[ x<4 \] The only prime numbers less than 4 are: \[ x\in\{2,3\} \] Now evaluate the GCD in both cases. If \[ x=2, \] then \[ \gcd(150,2)=2 \] which is prime. If \[ x=3, \] then \[ \gcd(150,3)=3 \] which is also prime. Thus, regardless of whether \(x=2\) or \(x=3\), the GCD is always a prime number. Therefore, the answer to the question is definitely Yes. Hence, the two statements together are sufficient. Final Answer: \[ \boxed{\text{Both statements (I) and (II) are sufficient}} \]