Direct Proofs

The following is the list of problems for Chapter 4 of the Book of Proof (page 100). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

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 \( a \vert b \) and \( a \vert c \) then \( a \vert (b+c) \).
7. Suppose \( a, b \in \mathbb{Z} \). If \( a \vert b \) then \( a^2 \vert b^2 \).
8. Suppose \( a \) is an integer. If \( 5 \vert 2a \), then \( 5 \vert a \).
9. Suppose \( a \) is an integer. If \( 7 \vert 4a \), then \( 7 \vert a \).
10. Suppose \( a \) and \( b \) are integers. If \( a \vert b \), then \( a \vert (3b^3-b^2+5b) \).
11. Suppose \( a, b, c, d, \in \mathbb{Z} \). If \( a \vert b \) and \( c \vert d \), then \( ac \vert bd \).
12. If \( x \in \mathbb{R} \) and \( 0 < x < 4 \), then \( \tfrac{4}{x(4-x)} \geq 1 \).
13. Suppose \( x, y \in \mathbb{R} \). If \( x^2 + 5y = y^2 + 5x \), then \( x=y \) or \( x+y = 5 \).
14. If \( n \in \mathbb{Z} \), then \( 5n^2 + 3n + 7 \) is odd. (Try cases.)
15. If \( n \in \mathbb{Z} \), then \( n^2 + 3n + 4 \) is even. (Try cases.)
16. If two integers have the same parity, then their sum is even. (Try cases.)
17. If two integers have opposite parity, then their product is even.
18. Suppose \( x \) and \( y \) are positive real numbers. If \( x < y \), then \( x^2 < y^2 \).
19. Suppose \( a, b \) and \( c \) are integers, If \( a^2 \vert b \) and \( b^3 \vert c \), then \( a^6 \vert c \).
20. If \( a \) is an integer and \( a^2 \vert a \), then \( a \in \{ -1,0,1 \} \).
21. If \( p \) is prime and \( k \) is an integer for which \( 0 < k < p \), then \( p \) divides \( \binom{p}{k} \).
22. If \( n \in \mathbb{N} \), then \( n^2 = 2 \binom{n}{2} + \binom{n}{1} \). (You may need a separate case for \( n=1 \).)
23. If \( n \in \mathbb{N} \), then \( \binom{2n}{n} \) is even.
24. If \( n \in \mathbb{N} \) and \( n \geq 2 \), then the numbers \( n!+2, n!+3, n!+4, n!+5, \dotsc, n!+n \) are all composite. (Thus, for any \( n\geq 2 \), one can find \( n \) consecutive composite numbers. This means there are arbitrarily larget “gaps” between prime numbers.)
25. If \( a, b, c, \in \mathbb{N} \) and \( c \leq b \leq a \), then \( \binom{a}{b} \binom{b}{c} = \binom{a}{b-c} \binom{a-b+c}{c} \).
26. Every odd integer is a difference of two squares. (Example \( 7 = 4^2-3^2 \), etc.)
27. Suppose \( a, b \in \mathbb{N} \). If \( \gcd(a,b)>1 \), then \( b \vert a \) or \( b \) is not prime.
28. If \( a, b, c \in \mathbb{Z} \), then \( c \cdot \gcd(a,b) \leq \gcd(ca,cb) \).