## 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** -> $$