Prove that there is no natural number that is even and odd

### Proof that no natural number can be both even and odd:

We will prove this statement by contradiction.

#### Definitions:
1. A **natural number** is a positive integer, i.e., \( \mathbb{N} = \{ 1, 2, 3, 4, \dots \} \).
2. A number is called **even** if it can be written in the form \( n = 2k \) for some integer \( k \).
3. A number is called **odd** if it can be written in the form \( n = 2k + 1 \) for some integer \( k \).

#### Step 1: Assume the opposite.
Suppose there exists a natural number \( n \) that is both even and odd.

#### Step 2: Express \( n \) as both even and odd.
- Since \( n \) is even, by definition, there must exist some integer \( k_1 \) such that:
  \[
  n = 2k_1
  \]
- Since \( n \) is also odd, by definition, there must exist some integer \( k_2 \) such that:
  \[
  n = 2k_2 + 1
  \]

#### Step 3: Equate both expressions for \( n \).
Since both expressions represent the same number \( n \), we can equate them:
\[
2k_1 = 2k_2 + 1
\]

#### Step 4: Simplify the equation.
Subtract \( 2k_2 \) from both sides:
\[
2k_1 - 2k_2 = 1
\]
Factor out the 2:
\[
2(k_1 - k_2) = 1
\]

#### Step 5: Reach a contradiction.
The left-hand side of the equation is an even number (because it is 2 times an integer), and the right-hand side is the number 1, which is odd. This is a contradiction because no even number can equal an odd number.

#### Step 6: Conclusion.
Since we have arrived at a contradiction, the assumption that a natural number can be both even and odd is false. Therefore, no natural number is both even and odd.

### Conclusion:
There is no natural number that is both even and odd.

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Here are some related questions:
1. What are the properties of even and odd numbers?
2. Can a sum of two odd numbers be even?
3. How can we determine whether large numbers are even or odd?
4. Is zero an even number, and why?
5. What are the properties of natural numbers in relation to even and odd classifications?

**Tip:** When adding or subtracting even and odd numbers, remember that:
- even + even = even,
- odd + odd = even,
- even + odd = odd.

Learn how to prove that no natural number can be both even and odd through a step-by-step contradiction proof. This problem focuses on number theory concepts, definitions of even and odd numbers, and logical contradiction, suitable for Grades 8-10.

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