Modulo and probability

Question

How can I prove that 4 modulo 5 is 4?

My though is floor of (4 / 5) is 0 then the remaining is = to the modulo.Am I right?

Answer 1

We have for every integer $a$ and modulo $m$ $$a\equiv a\pmod m$$

This is because $m\mid a-a$.

So, yes $4$ modulo $5$ is again $4$.

Answer 2

The 'modulo' operator gives you the remainder of the integer or Euclidean division between two positive integers. So, indeed, the result of the Euclidean division of 4 by 5 is 0, with remainder 4.

Answer 3

You proved it what your proved $\equiv \pmod n$ was an equivalence relationship.

$4 \equiv 4 \pmod 5$ because, being an equivalence relationship, equivalence modulo $n$ is reflexive. i.e. for all $a$, $a \equiv a \pmod n$.

Of course we had to prove equivalence modulo $n$ was an equivalence relationship in the first place.

Definition: for any $n \in \mathbb N$ and $a, b \in \mathbb Z$ we say $a \equiv b \pmod n$ if $n|a-b$ that is if $\frac {a-b}n$ is an integer.

Theorem: $\equiv \pmod n$ is an equivalence relationship. That is to say it is a) reflexive; b) symmetric; c) transitive.

Pf:

a) $\equiv \pmod n$ is reflexive. That is for all $a \in \mathbb Z$, $a \equiv a \pmod n$.

Pf of a) $a-a=0$. And $\frac 05 = 0$ so $5|a-a$ so $a\equiv a \pmod n$.

That's it, we're done. Your question is answered completely.

b) $\equiv \pmod n$ is symmetric. That is if $a \equiv b \pmod n$ then $b \equiv a \pmod n$.

Pf of b) $b-a = -(a-b)$ and $\frac {b-a}n = - \frac {a-b}n$ and if $\frac {a-b}n$ is an intger so is $-\frac {a-b}n$. So if $n|a-b$ then $n|b-a$ and if $a\equiv b\pmod n$ then $b\equiv a \pmod n$.

c) $\equiv \pmod n$ is transitive. That is if $a\equiv b\pmod n$ anc $b\equiv c \pmod n$ then $a \equiv c\pmod n$.

Pf of c) $\frac {a-c}n = \frac {a-b}n + \frac {b-c}n$. If $\frac {a-b}n$ and $\frac {b-c}n$ are integers, so is $\frac {a-c}n$. So if $n|a-b$ and $n|b-c$ then $n|(a-b) + (b-c) = a-c$. So if $a\equiv b\pmod n$ and $b\equiv c \pmod n$ then $a \equiv c \pmod n$.

So $\equiv pmod n$ is an equivalence relation as it is reflexive, symmetric, and transitive.