Motivation for learning Number theory

pages/content/ch4.mdx

@@ -1,6 +1,37 @@
 
+import { Callout, Tabs } from 'nextra/components'
+
 # Number Theory
 
+The Number Theory is a wonderful part of mathematics which studies how to solve various problems using properties that different numbers possess.
+
+## Motivation
+
+As a **motivation for studying**, the following example might be good:
+
+<Callout type="-"> 
+**Is there a solution of $x^3 + x^2 = y^4 + y + 1$ for some integers $x$, $y$?**
+
+Aaaand ... No there is no solution due to the fact that both sides can't be at the same time:
+
+$$
+\text{odd} = \text{odd} \text{ or } \text{even} = \text{even}
+$$ 
+
+**Let us take two cases:**
+1. **$y = \text{even}$**, then when we have $y^4 + y + 1$, then $y^4$ is as well even,. Now, adding  $(y^4) \  \text{even} + (y) \  \text{even} = \text{even}$. Finally, $((y^4 + y) + 1)$ results in $\text{even} + \text{odd}$, which is always $\text{odd}$.
+2. **$y = \text{odd}$**, then $y^4 + y + 1 = \text{odd} + \text{odd} + \text{odd} = \text{odd}$
+
+So the LHS is always odd. Now, considering LHS, we can do a **similar analysis**:
+
+1. **$x = \text{even}$**, then $x^3 + x^2 = \text{even} + \text{even} = \text{even}$
+2. **$x = \text{odd}$**, then $x^3 + x^2 = \text{odd} + \text{odd} = \text{even}$
+
+RHS is always $\text{even}$. Here we arrive at a **contradiction** ⚡️
+$\text{even} \neq \text{odd}$, which proves, that there is no solution to the given equation. 
+</Callout>
+
+This **example reveals** that by knowing how even and odd numbers react to adding them or raising to some power (in a sense **knowing some theory behind these numbers**), we can prove that there is no solution for a given equation.
 
 ## References
 
@@ -12,3 +43,5 @@ The following are some references to problem sets which were quite useful to get
 
 - [*Number Theory Document* - artofproblemsolving.com](https://artofproblemsolving.com/articles/files/SatoNT.pdf?srsltid=AfmBOopyi3E0r3pHpuDR7Oi9LV55TyzI33f5T_rF-udQyytDC7x8g6za) 
 
+
+

pages/content/ch4/division.mdx

@@ -19,6 +19,7 @@ import { Cards } from 'nextra/components'
   />
 </Cards>
 
+
 <Callout type="-"> 
 **Definition 4.1**: <br/>
 For integers $a$ and $b$, we say that **$a$ divides $b$**, denoted $a \mid b$, if there exists an integer $c$ such that $b = ac$. In this case, $a$ is called a **divisor** of $b$, and $b$ is called a **multiple** of $a$. If $a \neq 0$ and a divisor $c$ exists, it is called the **quotient** when $b$ is divided by $a$, and we write $$c = \frac{b}{a}$$ or $c = b/a$. 4. We write $a \nmid b$ if $a$ does not divide $b$.

pages/content/ch4/modular-arithmetic.mdx

@@ -7,3 +7,6 @@ import { Cards } from 'nextra/components'
 # Modular Arihmetic
 
 
+
+
+