Prove that x is an odd integer if and only if 5x − 1 is an even integer

To prove that \( x \) is an odd integer if and only if \( 5x - 1 \) is an even integer, we need to demonstrate both implications:

### 1. **If \( x \) is an odd integer, then \( 5x - 1 \) is an even integer.**

Assume \( x \) is an odd integer. By definition, an odd integer can be expressed as:
\[
x = 2k + 1
\]
for some integer \( k \).

Now, substitute this into the expression \( 5x - 1 \):
\[
5x - 1 = 5(2k + 1) - 1
\]
Expanding this:
\[
5x - 1 = 10k + 5 - 1 = 10k + 4
\]
Notice that \( 10k + 4 \) can be factored as:
\[
5x - 1 = 2(5k + 2)
\]
Since \( 5k + 2 \) is an integer, \( 5x - 1 \) is twice some integer, which means it is an even integer.

Thus, if \( x \) is an odd integer, \( 5x - 1 \) is indeed an even integer.

### 2. **If \( 5x - 1 \) is an even integer, then \( x \) is an odd integer.**

Assume \( 5x - 1 \) is an even integer. This means:
\[
5x - 1 = 2m
\]
for some integer \( m \).

To solve for \( x \), add 1 to both sides:
\[
5x = 2m + 1
\]
Now divide by 5:
\[
x = \frac{2m + 1}{5}
\]
Here, \( 2m + 1 \) is an odd integer (since the sum of an even and an odd number is odd), and an odd integer divided by 5 will result in another integer only if \( x \) is odd.

This implies that \( x \) must be of the form \( x = 2k + 1 \) for some integer \( k \), which means \( x \) is an odd integer.

Thus, if \( 5x - 1 \) is an even integer, \( x \) must be an odd integer.

### **Conclusion**

We have shown both directions:

1. If \( x \) is an odd integer, then \( 5x - 1 \) is an even integer.
2. If \( 5x - 1 \) is an even integer, then \( x \) is an odd integer.

Therefore, \( x \) is an odd integer **if and only if** \( 5x - 1 \) is an even integer.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What happens if \( x \) is an even integer in the expression \( 5x - 1 \)?
2. How would the proof change if the expression were \( 5x + 1 \) instead of \( 5x - 1 \)?
3. Can you prove that if \( x \) is even, then \( 5x - 1 \) is odd?
4. How does this proof apply to other linear functions of \( x \)?
5. Can you find another similar statement involving odd/even integers and a different linear combination?

**Tip:** Remember that even and odd integers have a predictable behavior when added, subtracted, or multiplied, which is helpful in proving statements involving parity.

Prove that x is an odd integer if and only if 5x − 1 is an even integer.

Explore a detailed proof that shows x is an odd integer if and only if 5x − 1 is an even integer. This problem involves key number theory concepts and is suitable for students in Grades 9-12. Learn how to apply the properties of odd and even numbers to solve logical equivalence problems.

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