Algebra Memorisation Points

$a|b$ means $b=ak$ for some $kϵz$. GCD of $a,bϵz$ is denoted $(a,b)$. Bézout’s Lemma: $(a,b)=ax+by$ for some $x,y\frac{ab}{(a,b)}=[a,b]$. Arithmetical Lemma A: $(a,b)=1$ (i.e. coprime) $⟺ax+by=1$ for some $x,yϵz$. Arithmetical Lemma B: $(a,b)=1$ and $a|bc⟹a|c$. Arithmetical Lemma C: $\left(\frac{a}{(a,b)},\frac{b}{(a,b)}\right)=1$. Arithmetical Lemma D: $a|bc⟹\frac{a}{(a,b)}|c$.

TARGET DECK MTH1001 Algebra

STARTI [Basic] Question: What does the notation $a|b$ mean? Back: $b=ak$ for some $kϵz$. ENDI STARTI [Basic] Question: How is the GCD of $a$ and $b$ denoted? Back: The GCD of $a,bϵz$ is denoted $(a,b)$. ENDI STARTI [Basic] Question: What is Bézout’s Lemma? Back: Bézout’s Lemma states that $(a,b)=ax+by$ for some $x,y$. Additionally, $\frac{ab}{(a,b)}=[a,b]$. ENDI STARTI [Basic] Question: What does Arithmetical Lemma A state about coprime integers $a$ and $b$? Back: $(a,b)=1$ (i.e. coprime) $⟺ax+by=1$ for some $x,yϵz$. ENDI STARTI [Basic] Question: What does Arithmetical Lemma B state regarding integers $a,b,c$ where $(a,b)=1$? Back: If $(a,b)=1$ and $a|bc$, then $a|c$. ENDI STARTI [Basic] Question: What is the relation between $\frac{a}{(a,b)}$ and $\frac{b}{(a,b)}$ as per Arithmetical Lemma C? Back: $\left(\frac{a}{(a,b)},\frac{b}{(a,b)}\right)=1$. ENDI STARTI [Basic] Question: What does Arithmetical Lemma D state about the relation between $a,b,$ and $c$? Back: If $a|bc$, then $\frac{a}{(a,b)}|c$. ENDI