91Ó°ÊÓ

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a \in \mathbb{Z}\). Then \(a^{2} \mid a\) if and only if \(a \in\\{-1,0,1\\}\)

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b \in \mathbb{Z}\). Prove that \(a+b\) is even if and only if \(a\) and \(b\) have the same parity.

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b \in \mathbb{Z} .\) If \(a b\) is odd, then \(a^{2}+b^{2}\) is even.

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). There is a prime number between 90 and 100 .

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). If \(n \in \mathbb{N},\) then \(2^{0}+2^{1}+2^{2}+2^{3}+2^{4}+\cdots+2^{n}=2^{n+1}-1\)

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Every real solution of \(x^{3}+x+3=0\) is irrational.

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). If \(n \in \mathbb{Z},\) then \(4 \mid n^{2}\) or \(4 \mid\left(n^{2}-1\right)\)

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b\) and \(c\) are integers. If \(a \mid b\) and \(a \mid\left(b^{2}-c\right),\) then \(a \mid c\).

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). If \(a \in \mathbb{Z},\) then \(4 \nmid\left(a^{2}-3\right)\)