Define Congruence. When we say that given integers a and b are congruent modulo n ? where n is fixed positive integer.

 Definition: Let $n$ be a fixed positive integer. Two integers $a$ and $b$ are said to be congruent modulo $n$}  denoted by $a \equiv b ~(mod ~n)$ if $n$ divides the difference $a - b$; that is, provided that $a - b = k \cdot n$ for some integer $k$. 

Now we give an equivalent  definition of congruent modulo $n$

 Definition: Let $n$ be a fixed positive integer. Two integers $a$ and $b$ are said to be \congruent modulo $n$,  denoted by $a  \equiv  b ~(mod~ n)$  if $b$ is a remainder when $a$ is  divided by $n$. 

Let $n = 5$, we can observe that $10  \equiv  0 ~(mod~ 5)$ Since $5$ divides $10-0= 10$. Similarly,  we can see that $21  \equiv  1 ~(mod~ 5)$ Since $5$ divides $21-1= 20$, $37  \equiv  2 ~(mod~ 5)$ Since $5$ divides $37-2= 35$, $43  \equiv  3 ~(mod~ 5)$ Since $5$ divides $43-3= 40$, $59  \equiv  4 ~(mod~ 5)$ Since $5$ divides $59-4= 55$.

When $ n \nmid (a - b)$, we say that $a$ is incongruent to $b$ modulo $n$, and in this case we write $a \not\equiv b ~(mod ~ n)$. For a simple example: $25 \not\equiv 12 ~(mod ~5)$, because $5$ does not divide  $25 - 12 = 13$.

   Given an integers $a$ and $n > 0$, when we divide $a$ by $n$   by Division Algorithm there exist  quotient $q$ and remainder $r$ such that $a = q \cdot n + r$ where $ 0 \leq r < n$ .

Then, by definition of congruence, $a \equiv r ~(mod ~n )$. Since there are $n$ choices for the remainder $r$, we see that every integer is congruent modulo $n$ to exactly one of the values 

$0, 1, 2, ... , n - 1$; in particular, $a \equiv 0 ~(mod ~n)$ if and only if $n ~|~a$. 

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