Class Journal Problems

--- tags: mth350, class-journal --- # Class Journal Problems ## Theorem 1.2.8 Let $a,b,c \in \mathbb{Z}$ such that $a = b + c$ with $a$ and $b$ not both zero. Then $\gcd(a,b) = \gcd(b,c)$. ## Theorem 1.2.9 (Bezout's Identity) For any integers $a$ and $b$ not both $0$, there are integers $x$ and $y$ such that $$ax + by = \gcd(a,b).$$ *Note*: This needs to be a *complete proof* that incorporates all the pieces we did in class, into one clear and self-contained argument. So you'll need to prove: - That $S = \{as + bt \, : \, s,t \in \mathbb{Z} \, \text{and} \ as + bt > 0\}$ has a smallest element - That the smallest of element of $S$ satisfies the definition of $\gcd(a,b)$. We spent most of class on February 1 proving one part of that definition. ## Further questions about Bezout's Identity - Prove or disprove: The weights $x,y$ that make $ax + by = \gcd(a,b)$ are unique. That is, there is only one possible pair of weights $x,y$ that make this equation true. - Prove or disprove the **converse of Theorem 1.2.9**: For any $a,b \in \mathbb{Z}$, if there exist integers $u,v$ such that $au + bv = N$, then $N = \gcd(a,b)$. - Prove that for any $a,b \in \mathbb{Z}$, if there exist integers $u,v$ such that $au + bv = 1$, then $\gcd(a,b) = 1$. This is the **converse of the Corollary to Theorem 1.2.9**; you may not use Theorem 1.2.9 in the proof! ## GCD's of separated integers This is a two-parter: 1. Prove that for any integer $a$, $\gcd(a, a+1) = 1$. 2. Prove that for any integer $a$, $\gcd(a, a+2)$ equals $1$ if $a$ is odd and equals $2$ if $a$ is even. *Note:* You may use the third bullet point in the previous problem (the converse to the Corollary of Theorem 1.2.9).