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

Step 1: Understanding the Question:
This time and work problem provides the completion time for a joint effort and for one individual, requiring us to calculate the solo completion time for the second individual.


Step 2: Key Formula or Approach:
An individual finishing a job in \(x\) days has a daily work rate of \(\frac{1}{x}\).
The joint work rate is the sum of individual rates: \(\text{Rate}(A+B) = \text{Rate}(A) + \text{Rate}(B)\).


Step 3: Detailed Explanation:
Since A and B together finish the job in 12 days, their combined daily work rate is:
\[ \text{Rate}(A+B) = \frac{1}{12} \] A working alone takes 20 days, meaning A's daily work rate is:
\[ \text{Rate}(A) = \frac{1}{20} \] Let B's daily work rate be \(\frac{1}{x}\), with \(x\) representing B's solo completion time.
\[ \text{Rate}(B) = \text{Rate}(A+B) - \text{Rate}(A) \] \[ \text{Rate}(B) = \frac{1}{12} - \frac{1}{20} \] Finding the least common multiple (LCM) of 12 and 20, which is 60, allows us to subtract the fractions:
\[ \text{Rate}(B) = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \] Because B completes \(\frac{1}{30}\) of the work per day, B will take 30 days to finish the job alone.


Step 4: Final Answer:
The correct choice is (B).