**Problem 3: Proof by Contraposition****Statement:**Prove the following by contraposition.If $ n^2 $ is not divisible by 3, then $ n $ is not divisible by 3.**Explanation:**To prove this statement using contraposition, we demonstrate that the contrapositive is true. The contrapositive of the given statement is: **If $ n $ is divisible by 3, then $ n^2 $ is divisible by 3.****Proof:**1. Assume that $ n $ is divisible by 3. This means there exists an integer $ k $ such that $ n = 3k $.2. Calculate $ n^2 $: \[ n^2 = (3k)^2 = 9k^2 = 3(3k^2) \]3. Since $ n^2 = 3(3k^2) $, $ n^2 $ is also divisible by 3.Thus, we have shown that if $ n $ is divisible by 3, then $ n^2 $ is divisible by 3, proving the contrapositive and therefore the original statement.

Statement:if A then B Contrapositive statement:if not B then not A

Answered: (3) Prove the following by…

Math

Advanced Math

(3) Prove the following by contraposition. If n? is not divisible by 3, then n is not divisible by 3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 3: Proof by Contraposition**

**Statement:**

Prove the following by contraposition.

If $ n^2 $ is not divisible by 3, then $ n $ is not divisible by 3.

**Explanation:**

To prove this statement using contraposition, we demonstrate that the contrapositive is true. The contrapositive of the given statement is:

**If $ n $ is divisible by 3, then $ n^2 $ is divisible by 3.**

**Proof:**

1. Assume that $ n $ is divisible by 3. This means there exists an integer $ k $ such that $ n = 3k $.

2. Calculate $ n^2 $:
\[
n^2 = (3k)^2 = 9k^2 = 3(3k^2)
\]

3. Since $ n^2 = 3(3k^2) $, $ n^2 $ is also divisible by 3.

Thus, we have shown that if $ n $ is divisible by 3, then $ n^2 $ is divisible by 3, proving the contrapositive and therefore the original statement.

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