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3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² + 4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x,y e R. Then x³ + x²y = y2 + xy if and only if y=x² or y=-x. Then (x + y)2 = x² + y² if and only if x = 0 or y = 0. 7. Suppose x,y e R. 8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a.

3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² + 4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x,y e R. Then x³ + x²y = y2 + xy if and only if y=x² or y=-x. Then (x + y)2 = x² + y² if and only if x = 0 or y = 0. 7. Suppose x,y e R. 8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a.

Advanced Engineering Mathematics
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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Please detailed prove on 3,5,9
**Exercises for Chapter 7** Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4–7. 1. Suppose \(x \in \mathbb{Z}\). Then \(x\) is even if and only if \(3x + 5\) is odd. 2. Suppose \(x \in \mathbb{Z}\). Then \(x\) is odd if and only if \(3x + 6\) is odd. 3. Given an integer \(a\), then \(a^3 + a^2 + a\) is even if and only if \(a\) is even. 4. Given an integer \(a\), then \(a^2 + 4a + 5\) is odd if and only if \(a\) is even. 5. An integer \(a\) is odd if and only if \(a^3\) is odd. 6. Suppose \(x, y \in \mathbb{R}\). Then \(x^3 + x^2 y - y^2 + x y\) if and only if \(y = x^2\) or \(y = -x\). 7. Suppose \(x, y \in \mathbb{R}\). Then \((x+y)^2 = x^2 + y^2\) if and only if \(x = 0\) or \(y = 0\). 8. Suppose \(a, b \in \mathbb{Z}\). Prove that \(a \equiv b \pmod{10}\) if and only if \(a \equiv b \pmod{2}\) and \(a \equiv b \pmod{5}\). 9. Suppose \(a \equiv \mathbb{Z}\). Prove that \(14 \mid a\) if and only if \(7 \mid a\) and \(2 \mid a\). 10. If \(a \equiv \mathbb{Z}\), then \(a^3 \equiv a \pmod{3}\). 11. Suppose \(a, b \equiv \mathbb{Z}\). Prove that \((a - 3)b^2\) is even if and only if \(a\) is odd or \(b\) is even. 12. There exist a positive real number \(
Transcribed Image Text:**Exercises for Chapter 7** Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4–7. 1. Suppose \(x \in \mathbb{Z}\). Then \(x\) is even if and only if \(3x + 5\) is odd. 2. Suppose \(x \in \mathbb{Z}\). Then \(x\) is odd if and only if \(3x + 6\) is odd. 3. Given an integer \(a\), then \(a^3 + a^2 + a\) is even if and only if \(a\) is even. 4. Given an integer \(a\), then \(a^2 + 4a + 5\) is odd if and only if \(a\) is even. 5. An integer \(a\) is odd if and only if \(a^3\) is odd. 6. Suppose \(x, y \in \mathbb{R}\). Then \(x^3 + x^2 y - y^2 + x y\) if and only if \(y = x^2\) or \(y = -x\). 7. Suppose \(x, y \in \mathbb{R}\). Then \((x+y)^2 = x^2 + y^2\) if and only if \(x = 0\) or \(y = 0\). 8. Suppose \(a, b \in \mathbb{Z}\). Prove that \(a \equiv b \pmod{10}\) if and only if \(a \equiv b \pmod{2}\) and \(a \equiv b \pmod{5}\). 9. Suppose \(a \equiv \mathbb{Z}\). Prove that \(14 \mid a\) if and only if \(7 \mid a\) and \(2 \mid a\). 10. If \(a \equiv \mathbb{Z}\), then \(a^3 \equiv a \pmod{3}\). 11. Suppose \(a, b \equiv \mathbb{Z}\). Prove that \((a - 3)b^2\) is even if and only if \(a\) is odd or \(b\) is even. 12. There exist a positive real number \(
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