Analyzing Sufficiency for Natural Numbers

The question asks us to determine which of the natural numbers X, Y, Z, U, and V are even. We are given two statements and need to decide if either or both are sufficient to answer this question.

Let's analyze each statement separately and then together.

Analyzing Statement 1: X, Y, Z, U, and V are consecutive numbers

This statement tells us that the numbers follow each other in order, with a difference of 1 between each consecutive pair. For example, they could be 1, 2, 3, 4, 5 or 2, 3, 4, 5, 6, or 10, 11, 12, 13, 14, and so on.

If the sequence starts with an odd number, the pattern of parity (even or odd) would be: Odd, Even, Odd, Even, Odd.

If the sequence starts with an even number, the pattern of parity would be: Even, Odd, Even, Odd, Even.

Statement 1 alone does not provide information about whether the first number (X) is even or odd. Therefore, we cannot definitively determine which of the numbers X, Y, Z, U, and V are even. For example, if X=1, then Y=2, Z=3, U=4, V=5 (Y and U are even). But if X=2, then Y=3, Z=4, U=5, V=6 (X, Z, and V are even). The set of even numbers changes depending on the starting number.

Thus, Statement 1 alone is insufficient to answer the question.

Analyzing Statement 2: Z is an odd number

This statement gives us specific information about the parity of one number, Z. However, we do not know the relationship between Z and the other numbers (X, Y, U, V) based on this statement alone. They could be any five natural numbers where Z happens to be odd.

For instance, Z could be 3, and the other numbers could be 1, 2, 4, 5 (X=1, Y=2, U=4, V=5, Z=3). Here Y and U are even. But Z could be 3, and the other numbers could be 6, 7, 8, 9 (X=6, Y=7, U=8, V=9, Z=3). Here X and U are even. We cannot determine which of X, Y, U, and V are even based on Z being odd alone.

Thus, Statement 2 alone is insufficient to answer the question.

Analyzing Statements 1 and 2 Together

Now, let's consider both statements together:

1. X, Y, Z, U, and V are consecutive numbers.
2. Z is an odd number.

Since X, Y, Z, U, and V are consecutive natural numbers, their parities must alternate. The sequence is in increasing order. Z is the middle number in the sequence X, Y, Z, U, V.

The sequence of numbers is ... (Z-2), (Z-1), Z, (Z+1), (Z+2) ... which corresponds to X, Y, Z, U, V.

• X = Z - 2
• Y = Z - 1
• Z = Z
• U = Z + 1
• V = Z + 2

We are given that Z is an odd number. Let's determine the parity of the other numbers based on Z being odd:

• If Z is Odd, then Z - 1 (which is Y) must be Even.
• If Z is Odd, then Z - 2 (which is X) must be Odd.
• Z is given as Odd.
• If Z is Odd, then Z + 1 (which is U) must be Even.
• If Z is Odd, then Z + 2 (which is V) must be Odd.

So, the parity sequence is: Odd (X), Even (Y), Odd (Z), Even (U), Odd (V).

Using both statements, we can definitively identify which numbers are even: Y and U are the even numbers in the sequence.

Therefore, both statements together are sufficient to answer the question.

Conclusion on Sufficiency

Based on our analysis, Statement 1 alone is insufficient, Statement 2 alone is insufficient, but both statements 1 and 2 together are sufficient to determine which of the numbers are even.

This aligns with Option 1: Both statement 1 and 2 together are sufficient.

Revision Table: Data Sufficiency Concepts

When tackling Data Sufficiency problems involving consecutive numbers and parity, remember the following key concepts:

• Consecutive Numbers: These are numbers that follow each other in order, differing by 1. Their parities always alternate (Odd, Even, Odd, Even...).
• Parity: Whether a number is even or odd. An even number is divisible by 2, an odd number is not.
• Alternating Parity: In a sequence of consecutive integers, if one number is odd, the next must be even, and the one before must be even. If one number is even, the next must be odd, and the one before must be odd.
• Sufficiency: A statement or set of statements is sufficient if the information provided allows you to definitively answer the question asked. If you can find multiple possible answers using the information, it is insufficient.

Additional Information: Properties of Natural Numbers

Natural numbers are positive integers (1, 2, 3, ...). Understanding their properties is crucial for number-based questions.

• Set of Natural Numbers: $\{1, 2, 3, 4, 5, \dots\}$. Some definitions also include 0. In this problem, the context implies positive integers greater than 0.
• Parity Rules:
  • Even + Even = Even
  • Odd + Odd = Even
  • Even + Odd = Odd
  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even
• Consecutive Integer Sequences: In any sequence of 'n' consecutive integers:
  • If n is odd, there is a clear middle term. The parities alternate around the middle term.
  • If n is even, there is no single middle term, but the parities still alternate.
  • There is always at least one even and one odd number if the sequence length is > 1.
  • In any 5 consecutive integers, there will be either 3 odd and 2 even, or 2 odd and 3 even numbers.

In this problem, having 5 consecutive numbers and knowing the parity of the middle one (Z) was sufficient because it fixed the alternating pattern, allowing us to deduce the parity of all other numbers (X, Y, U, V).