Maths Metropolis

Introduction We have previously covered a short course on divisibility. In this chapter, we will generally focus on congruences. Loosely, congruence is a short notation that is extremely useful in solving various problems in number theory. Precisely, it is much more than just a notation.\\ Without further ado, let's get straight to the punchline. Congruence Notation If $a, b, m \in \mathbb{Z}$ such that $m \mid a - b$, then we write this as $a \equiv b \pmod{m}$. This is really all there is to it! But with this notation, we will prove several interesting results in an easy way. This notation or relation $\equiv$ has many similarities with the "$=$" sign. Some of the similar properties are enumerated below:   If $a \equiv b \pmod{m}$ and $b \equiv c \pmod{m}$, then $a \equiv c \pmod{m}$.   If $a \equiv b \pmod{m}$, then this is equivalent to $b \equiv a \pmod{m}$ and $a - b \equiv 0 \pmod{m}$.    If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $a + c \equiv b...

Combinatorics is a branch of mathematics that is mainly concerned with counting. This branch gives us a reason to inspect the art of counting in more detail and it further proves to its readers that it's rewarding as well!  In fact many of you might be familiar with the fundamental rule of counting. To put it more simply, you use this rule in your everyday life.  But before jumping into what the rule is, and  the technicalities, consider the figure below, We want to find the number of ways, in which we can travel from point $A$ to point $C$ via $B.$ In order to reach $B$ from $A$ we have $3$ options: to either travel by Ship $1,5$ and Boat $2.$ Finally, on reaching point $B$ there are again $2$ ways of moving to $C$ using either Boat $4$ or Ship $3.$ So, the total number of ways in which we can go to $C$ from $A$ via $B$ are as follows: Ship $5$ and then Ship $3$  Ship $5$ and then Boat $4$  Ship $1$ and then Ship $3$  Ship $1$ and then Boat $4$  Boat ...