Question

Is the number a prime number?
1. The number is divisible by a prime factor.
2. The number is positive.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

For a statement to be sufficient, it must lead to the same answer (always 'yes' or always 'no') in every possible case. If you can find one example that gives a 'yes' and another that gives a 'no', the statement is insufficient. This is the "test with different cases" strategy.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The question asks whether a given number is prime. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. We need to check if the statements provide enough information to give a definitive 'yes' or 'no' answer.
Step 2: Detailed Explanation:
Let the number be \( n \).
Analyzing Statement (1): The number is divisible by a prime factor.
By the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers. This means every integer greater than 1 has at least one prime factor.
- Case 1 (Yes): Let \( n = 7 \). 7 is divisible by the prime factor 7. The number is prime.
- Case 2 (No): Let \( n = 6 \). 6 is divisible by the prime factors 2 and 3. The number is not prime (it's composite).
Since we can find examples where the answer is 'yes' and examples where the answer is 'no', statement (1) alone is not sufficient.
Analyzing Statement (2): The number is positive.
This tells us that \( n>0 \).
- Case 1 (Yes): The number could be 5, which is positive and prime.
- Case 2 (No): The number could be 4, which is positive but not prime.
- Case 3 (No): The number could be 1, which is positive but not prime.
This statement also fails to give a definitive answer. Thus, statement (2) alone is not sufficient.
Analyzing Statements (1) and (2) Together:
Combining the statements, we know the number is positive and is divisible by a prime factor. This essentially describes any integer greater than 1.
- Case 1 (Yes): Let \( n = 3 \). It is positive and divisible by the prime factor 3. The number is prime.
- Case 2 (No): Let \( n = 9 \). It is positive and divisible by the prime factor 3. The number is not prime.
Even with both statements, we cannot determine if the number is prime.
Step 3: Final Answer:
Since both statements together are not enough to answer the question, additional data is needed.