Prove by contrapositive$$  $$, if 8 x (n^2 -1), then n is even

To prove the statement "If $ 8 \mid (n^2 - 1) $, then $ n $ is even" by **contrapositive**, we will instead prove the contrapositive of this statement. 

The contrapositive of the given statement is:

"If $ n $ is odd, then $ 8 \nmid (n^2 - 1) $."

### Steps for the proof by contrapositive:

1. **Assume $ n $ is odd**.  
   - This means $ n = 2k + 1 $ for some integer $ k $, where $ k $ is an integer.
  
2. **Express $ n^2 - 1 $** in terms of $ k $:  
   \[
   n^2 = (2k + 1)^2 = 4k^2 + 4k + 1
   \]
   Therefore,
   \[
   n^2 - 1 = (4k^2 + 4k + 1) - 1 = 4k^2 + 4k = 4(k^2 + k)
   \]
   So, $ n^2 - 1 = 4(k^2 + k) $.

3. **Check divisibility by 8**:  
   Since $ n^2 - 1 = 4(k^2 + k) $, the expression is divisible by 4, but we need to check if it is divisible by 8.

   - For $ n^2 - 1 $ to be divisible by 8, $ k^2 + k $ must be divisible by 2 (i.e., it must be an even number).

4. **Analyze $ k^2 + k $**:  
   - Notice that $ k^2 + k = k(k + 1) $. This is a product of two consecutive integers, one of which is always even.
   - Therefore, $ k(k + 1) $ is always divisible by 2, but not necessarily by 4.

   - For example, if $ k = 1 $, $ k^2 + k = 1(1 + 1) = 2 $, which is divisible by 2 but not by 4.

5. **Conclusion**:  
   Since $ n^2 - 1 = 4(k^2 + k) $ is divisible by 4 but not necessarily by 8 when $ n $ is odd, this shows that if $ n $ is odd, then $ n^2 - 1 $ is not divisible by 8.

Thus, we have proved the contrapositive: **If $ n $ is odd, then $ 8 \nmid (n^2 - 1) $**.

Since the contrapositive is true, the original statement "If $ 8 \mid (n^2 - 1) $, then $ n $ is even" is also true by the logic of contrapositive.

---

Would you like more details on any part of this proof?

Here are 5 related questions to expand on this topic:
1. What are the other methods to prove this type of statement (direct proof, proof by contradiction)?
2. How does divisibility work in modular arithmetic for odd and even numbers?
3. Can we prove that $ n^2 - 1 $ is divisible by a higher power of 2 for certain values of $ n $?
4. What other properties of odd and even numbers are useful in proofs of divisibility?
5. How can we apply the contrapositive method to more complex divisibility problems?

**Tip:** In modular arithmetic, recognizing patterns for odd and even numbers can simplify many divisibility problems.

Prove by contrapositive, if 8 | (n^2 - 1), then n is even.

Explore how to prove by contrapositive that if 8 divides (n^2 - 1), then n must be even. This proof covers concepts in number theory, divisibility, and uses a detailed breakdown of the contrapositive method. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation