1. Suppose x € Z. Then x is even if and only if 3x +5 is odd. 2. Suppose x € Z. Then x is odd if and only if 3x + 6 is odd. 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 € R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. 7. Suppose x, y € R. Then (x + y)² = x² + y2 if and only if x = 0 or y = 0. 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. 10. If a € Z, then a³ = a (mod 3).
1. Suppose x € Z. Then x is even if and only if 3x +5 is odd. 2. Suppose x € Z. Then x is odd if and only if 3x + 6 is odd. 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 € R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. 7. Suppose x, y € R. Then (x + y)² = x² + y2 if and only if x = 0 or y = 0. 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. 10. If a € Z, then a³ = a (mod 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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Prove the following statements
ANSWER #6 ONLY
Transcribed Image Text:1. Suppose x € Z. Then x is even if and only if 3x +5 is odd.
2. Suppose x € Z. Then x is odd if and only if 3x + 6 is odd.
3. Given an integer a, then a³ + a² +a is even if and only if a is ever
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 € R. Then x³+x²y = y² + xy if and only if y=x² or y=-x.
7. Suppose x, y € R. Then (x + y)² = x² + y² if and only if x = 0 or y = 0.
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.
10. If a € Z, then a³ = a (mod 3).
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