Understanding the Data Sufficiency Question

The question asks us to determine which of the five natural numbers, A, B, C, D, and E, are odd. To answer this question, we need to know the parity (whether it's odd or even) of each of these five numbers. We are given two statements and need to assess if either statement alone, or both statements together, provide enough information to determine the odd numbers among A, B, C, D, and E.

Analyzing Statement 1: Natural Numbers

Statement 1 says: A, B, C, D and E are natural numbers.

• Natural numbers are the counting numbers: 1, 2, 3, 4, 5, and so on.
• Natural numbers include both odd numbers (1, 3, 5, ...) and even numbers (2, 4, 6, ...).
• This statement only tells us the set these numbers belong to, but it gives absolutely no information about the specific values or the parity (odd or even) of A, B, C, D, or E.
• For example, A could be 1 (odd) or 2 (even). B could be 3 (odd) or 4 (even), and so on for all the numbers.

Since Statement 1 does not allow us to determine whether A, B, C, D, or E are odd or even, Statement 1 alone is not sufficient to answer the question.

Analyzing Statement 2: B is a Prime Number

Statement 2 says: B is a prime number.

• A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
• Examples of prime numbers are 2, 3, 5, 7, 11, 13, ...
• Let's look at the parity of prime numbers:
  • The number 2 is a prime number, and it is an even number.
  • The number 3 is a prime number, and it is an odd number.
  • The number 5 is a prime number, and it is an odd number.
  • In fact, 2 is the only even prime number. All other prime numbers are odd.
• Statement 2 tells us that B is a prime number. This means B could be 2 (even) or it could be any other prime number like 3, 5, 7, etc. (all odd). We cannot definitively say whether B is odd or even based solely on Statement 2.
• Furthermore, Statement 2 provides no information whatsoever about the numbers A, C, D, or E. They could be any natural numbers, odd or even.

Since Statement 2 does not allow us to determine whether B is odd or even, and provides no information about the other numbers, Statement 2 alone is not sufficient to answer the question.

Analyzing Combined Statements 1 and 2

Now, let's consider both statements together:

1. A, B, C, D and E are natural numbers.
2. B is a prime number.

• Statement 1 establishes that the numbers belong to the set of natural numbers. This is consistent with Statement 2, as prime numbers are a subset of natural numbers (excluding 1).
• Statement 2 still states that B is a prime number. As discussed earlier, a prime number can be the number 2 (which is even) or any other prime (which are all odd).
• Even with both statements, we cannot definitively say whether B is odd or even. It depends on which specific prime number B is.
• Also, combining the statements gives us no additional information about A, C, D, or E beyond them being natural numbers. They could be any combination of odd or even numbers.

Therefore, even combining both Statement 1 and Statement 2 does not provide enough information to determine which of the natural numbers A, B, C, D, and E are odd numbers.

Conclusion on Sufficiency

Based on our analysis:

• Statement 1 alone is not sufficient.
• Statement 2 alone is not sufficient.
• Both statements together are not sufficient.

We cannot answer the question using the information provided in either statement or both statements combined.

Revision Table: Key Concepts Review

Additional Information: Parity and Prime Numbers

The concept of parity (being either odd or even) is fundamental in number theory. A natural number is either odd or even; there is no other possibility. Knowing the parity of a number gives specific information about its properties and divisibility.

Prime numbers are a special set within natural numbers. Their definition is based on their divisors. The fact that 2 is the only even prime number is a crucial detail often tested. All other prime numbers must be odd, because if an odd number greater than 1 had any even divisor other than 2, it would also have 2 as a divisor, meaning it wouldn't be prime (unless it was 2 itself, but 2 is even). If an odd number had only odd divisors other than 1 and itself, it would fit the definition of an odd prime.

In Data Sufficiency problems, it's important to use only the information given in the statements and common mathematical definitions. Do not assume any additional properties or relationships between the variables unless explicitly stated.