Prove the following by contradiction: Proposition: Suppose \(n\) is an integer. If \(n\) is odd, then \(n^2\) is odd.
Consider the following proposition:
Proposition: For every \(n \in \mathbb{Z}, 4 \not\mid (n^2 + 2)\).
Set up the proof by contradiction. In other words, write the first line of the proof, beginning with “Suppose…”.
Complete the proof by contradiction of the proposition from #2.
Prove the following proposition: Proposition: Suppose \(a, b \in \mathbb{Z}\). If \(4 \mid (a^2 + b^2)\), then \(a\) and \(b\) are not both odd.
Definition: An integer \(n\) is even if \(n = 2a\) for some integer \(a \in \mathbb{Z}\).
Definition: An integer \(n\) is odd if \(n = 2a + 1\) for some integer \(a \in \mathbb{Z}\).
Definition: Suppose \(a\) and \(b\) are integers. We say that \(a\) divides \(b\), written \(a \mid b\), if \(b = ac\) for some \(c \in \mathbb{Z}\). In this case we also say that \(a\) is a divisor of \(b\), and that \(b\) is a multiple of \(a\).
Definition: Two integers have the same parity if they are both even or they are both odd. Otherwise they have opposite parity.
Definition: Given integers \(a\) and \(b\) and \(n \in \mathbb{Z}\), we say that \(a\) and \(b\) are congruent modulo \(\mathbf{n}\) if \(n \mid (a - b)\). We write this as \(a \equiv b \pmod{n}\). If \(a\) and \(b\) are not congruent modulo \(n\), then we write \(a \not\equiv b \pmod{n}\).
Definition/reminder: A real number \(x\) is rational if \(x = \frac{a}{b}\) for some integers \(a, b\). A number that is not rational is irrational.