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n2 - 2 is

n2 - 2 is

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Problem Statement:** Use contradiction to prove that there is no integer \( n \) such that \( n^2 - 2 \) is divisible by 4. **Solution Outline:** 1. **Assumption for Contradiction:** - Assume that there exists an integer \( n \) such that \( n^2 - 2 \equiv 0 \pmod{4} \). This means \( n^2 \equiv 2 \pmod{4} \). 2. **Analyzing Possible Values of \( n \):** - Consider all possible integer residues of \( n \) modulo 4: \( n \equiv 0, 1, 2, 3 \pmod{4} \). 3. **Case Analysis:** - **Case 1:** \( n \equiv 0 \pmod{4} \) - Then \( n^2 \equiv 0^2 \equiv 0 \pmod{4} \). - **Case 2:** \( n \equiv 1 \pmod{4} \) - Then \( n^2 \equiv 1^2 \equiv 1 \pmod{4} \). - **Case 3:** \( n \equiv 2 \pmod{4} \) - Then \( n^2 \equiv 2^2 \equiv 4 \equiv 0 \pmod{4} \). - **Case 4:** \( n \equiv 3 \pmod{4} \) - Then \( n^2 \equiv 3^2 \equiv 9 \equiv 1 \pmod{4} \). 4. **Conclusion:** - None of the possible values for \( n \) satisfy \( n^2 \equiv 2 \pmod{4} \). - Hence, the assumption that such an integer \( n \) exists is false. This completes the proof by contradiction that there is no integer \( n \) such that \( n^2 - 2 \) is divisible by 4.
Transcribed Image Text:**Problem Statement:** Use contradiction to prove that there is no integer \( n \) such that \( n^2 - 2 \) is divisible by 4. **Solution Outline:** 1. **Assumption for Contradiction:** - Assume that there exists an integer \( n \) such that \( n^2 - 2 \equiv 0 \pmod{4} \). This means \( n^2 \equiv 2 \pmod{4} \). 2. **Analyzing Possible Values of \( n \):** - Consider all possible integer residues of \( n \) modulo 4: \( n \equiv 0, 1, 2, 3 \pmod{4} \). 3. **Case Analysis:** - **Case 1:** \( n \equiv 0 \pmod{4} \) - Then \( n^2 \equiv 0^2 \equiv 0 \pmod{4} \). - **Case 2:** \( n \equiv 1 \pmod{4} \) - Then \( n^2 \equiv 1^2 \equiv 1 \pmod{4} \). - **Case 3:** \( n \equiv 2 \pmod{4} \) - Then \( n^2 \equiv 2^2 \equiv 4 \equiv 0 \pmod{4} \). - **Case 4:** \( n \equiv 3 \pmod{4} \) - Then \( n^2 \equiv 3^2 \equiv 9 \equiv 1 \pmod{4} \). 4. **Conclusion:** - None of the possible values for \( n \) satisfy \( n^2 \equiv 2 \pmod{4} \). - Hence, the assumption that such an integer \( n \) exists is false. This completes the proof by contradiction that there is no integer \( n \) such that \( n^2 - 2 \) is divisible by 4.
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