update notations and fix typos

pages/Swap/Math4351_L1.md

@@ -48,7 +48,7 @@ Since $r'=b(q-q')+r \geq b(q-q') \geq b$, which contradicts that $r' < b$.
 
 Therefore, $q=q'$ and $r=r'$.
 
-EOP
+QED
 
 #### Definition: Divisibility
 
@@ -74,7 +74,7 @@ Some proof examples:
 
 (2) Since $a \mid b$ and $b \mid c$, there exist $k, l \in \mathbb{Z}$ such that $b = ak$ and $c = bl$. Then $c = bl = (ak)l = a(kl)$, so $a \mid c$.
 
-EOP
+QED
 
 (3) If $a \mid b$ and $b \mid a$, then there exist $k, l \in \mathbb{Z}$ such that $b = ak$ and $a = bl$. Then $a = bl = (ak)l = a(kl)$, so $a(1-kl) = 0$.
 
@@ -82,7 +82,7 @@ Case 1: $a=0$, then $b=0$, so $a=b$.
 
 Case 2: $a \neq 0$, then $1-kl=0$, so $kl=1$. Since $k, l \in \mathbb{Z}$, $k=l=\pm 1$, so $a = \pm b$.
 
-EOP
+QED
 
 #### Definition: Divisor
 
@@ -142,7 +142,7 @@ By property of divisibility (4), $d \mid bk + (a-bk) = a$.
 
 Therefore, $d \in D(a) \cap D(b)$.
 
-EOP
+QED
 
 This theorem gives rise to the Euclidean algorithm which is a efficient way to compute the greatest common divisor of two integers. $2\Theta(\log n)+1=O(\log n)$ ([Proof in CSE 442T Lecture 7](https://notenextra.trance-0.com/CSE442T/CSE442T_L7#euclidean-algorithm)).