91Ó°ÊÓ

Determine which of the following subsets of \(\mathbb{R}\) are well-ordered: (a) \\{\\} (b) \\{-9,-7,-3,5,11\\} (c) \\{0\\}\(\cup \mathbb{Q}^{+}\) (d) \(2 \mathbb{Z}\) (e) \(5 \mathbb{N}\) (f) \(\\{-6,-5,-4, \ldots\\}\)

Let \(a, b, c,\) and \(d\) be integers with \(a, c \neq 0 .\) Prove that (a) If \(a \mid b\) and \(c \mid d\), then \(a c \mid b d\). (b) If \(a c \mid b c,\) then \(a \mid b\).

Use induction to prove that for any integer \(n \geq 2,\) if \(a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{Z}\) and \(p\) is a prime such that \(p \mid a_{1} a_{2} \cdots a_{n},\) then \(p \mid a_{i}\) for some \(i,\) where \(1 \leq i \leq n\)

What are the possible values of \(\operatorname{gcd}(2 a+5 b, 5 a-2 b)\) if the two positive integers \(a\) and \(b\) are relatively prime?

Let \(a, b,\) and \(c\) be integers such that \(a, b \neq 0 .\) Prove that if \(a \mid b\) and \(b \mid c,\) then \(a \mid c\).

Prove that $$ b \bmod a \in\\{0,1,2, \ldots,|a|-1\\} $$ for any integers \(a\) and \(b\), where \(a \neq 0\).

Richard follows a very rigid routine. He orders a pizza for lunch every 10 days, and has dinner with his parents every 25 days. If he orders a pizza for lunch and has dinner with his parents today, when will he do both on the same day again?

Find the sum and product of 1053 and 1761 in \(\mathbb{Z}_{17}\).

Prove that \(\sqrt{p}\) is irrational for any prime number \(p\).

Compute \(\operatorname{gcd}(15 \cdot 50,25 \cdot 21),\) and \(\operatorname{lcm}(15 \cdot 50,25 \cdot 21)\)