Question 8 please write out proof, assume , therefor etc **Exercises for Chapter 4***Use the method of direct proof to prove the following statements:*1. If \( x \) is an even integer, then \( x^2 \) is even.2. If \( x \) is an odd integer, then \( x^3 \) is odd.3. If \( a \) is an odd integer, then \( a^2 + 3a + 5 \) is odd.4. Suppose \( x, y \in \mathbb{Z} \). If \( x \) and \( y \) are odd, then \( xy \) is odd.5. Suppose \( x, y \in \mathbb{Z} \). If \( x \) is even, then \( xy \) is even.6. Suppose \( a, b, c \in \mathbb{Z} \). If \( a \mid b \) and \( a \mid c \), then \( a \mid (b + c) \).7. Suppose \( a, b \in \mathbb{Z} \). If \( a \mid b \), then \( a^2 \mid b^2 \).8. Suppose \( a \) is an integer. If \( 5 \mid 2a \), then \( 5 \mid a \).9. Suppose \( a \) is an integer. If \( 7 \mid 4a \), then \( 7 \mid a \).10. Suppose \( a \) and \( b \) are integers. If \( a \mid b \), then \( a \mid (3b^3) \).11. Suppose \( a, b, c, d \in \mathbb{Z} \). If \( a \mid b \) and \( c \mid d \), then \( ac \mid bd \).12. If \( x \in \mathbb{R} \) and \( 0 < x < 4 \), then \(\frac{4}{x(4-x)} \geq 1\).13. Suppose \( x, y \in \mathbb{R} \). If \( x^2 + 5y = y^2 + 5x \), then \( x = y \) or \( x + y = 5 \).14. If \( n \in \mathbb

Name: Question 8 please write out proof, assume , therefor etc **Exercises f
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Question 8 please write out proof, assume , therefor etc **Exercises for Chapter 4***Use the method of direct proof to prove the following statements:*1. If \( x \) is an even integer, then \( x^2 \) is even.2. If \( x \) is an odd integer, then \( x