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10 Courses/01 - WiSe 2025_26/DAS/Arithmetic.md

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+## Asymptotic Equivalence Classes (Big-O)
+
+The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
+
+$f asymp g <==> f in O(g) and g in O(f)$
+
+Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
+
+###  The "Dominant Term" Rule
+To find which class a function belongs to, simplify it to its core growth rate:
+1.  **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
+2.  **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
+3.  **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
+
+## Euclidian Algorithm
+Purpose is to find the **GCD** (Greatest Common Divisor).
+
+### Core Rule
+Fill out this formula:
+$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
+1. **Divide** the bigger number by the smaller one
+2. **How many times** does it fit -> $"Quotient"$
+3. Find out **whats leftover** -> $"Remainder"$
+4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
+
+Result is the last $"Remainder"$ that is not $0$.
+
+### Bezout Coefficients
+**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
+
+1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
+2.  **Substitute remainders** 
+3. **Simplify** (Final Form: $"GCD" = x * "Dividend" + "y * Divisor"$)
+
+### Extended Euclidian Algorithm
+Only valid for GCD = 1
+
+1. **Find GCD** using Euclidian Algorithm
+	1. $A = q_1 * B + r_1$
+	2. $B = q_2 * r_1 + r_2$
+	3. $r_1 = q_3 * r_2 + 1$
+
+2. **Unwrap**
+	1. $1 = r_1 - (q_3 * r_2)$ mit $r_2 = B - q_2 * r_1$
+	2. $1 = r_1 - (q_3 * (B - q_2 * r_1))$ mit $q_3$ ausmultiplizieren
+	3. $1 = (1 + q_3 * q_2) * r_1 - q_3 * B$  mit $r_1 = A - q_1 * B$
+	4. $1 = (1 + q_3 * q_2) * (A - q_1 * B) - q_3 * B$
+
+3. **Simplify to** $x * A + y * B$
+4. Coefficient of $A$ is the modular inverse
+
+## Totient function
+Denoted by $phi(n) = n (1 - 1/p_1)(1 - 1/p_2)...(1-1/p_k)$ where $p_i$ is a prime factor of $n$.
+
+