(3) Prove the following by contraposition. If n? is not divisible by 3, then n is not divisible by 3.
(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
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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a635735-95d3-44e8-ac8f-77d69c434691%2Ffede18fe-4188-40e3-88fc-931fb6b7c0e7%2Fwn7pgn_processed.png&w=3840&q=75)
Transcribed Image Text:**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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