Visual Proof for Congruence Relation

Question

Can anybody please provide a convincing visual proof that

$a ≡ b \pmod {n}$ if $a-b$ is a multiple of $n$

is equivalent to

$a ≡ b \pmod n$ if $a$ and $b$ have the same remainder modulo $n$?

Edit: I am thinking of number lines, coloured squares, circle sectors etc. The symbolic proof would be helpful too, but I want to understand this in a more concrete way. Having the connections made between the symbolic proof and a more visual one would be ideal.

Answer 1

Mark an integer $r$ between $0$ and $n-1$ on the number line. Think of that as the remainder for hypothetical divisions you haven't done yet.

Now all the numbers that leave that remainder when divided by $n$ are spaced $n$ apart starting at $r$: $$ \ldots , r-2n, r-n, r, r+n, r+2n, r + 3n , \ldots $$ It's clear that any two of these are a multiple of $n$ apart.

Answer 2

It is obvious.

Simply write the Euclidean divisions: $ a=qn+r,\ b=q'n+r'$, $\: 0\leq r,r'<n$, so $a-b=(q-q')n+(r-r')$ and observe that $$-n<r-r'<n, \enspace\text{i.e}\quad |r-r'| <n.$$

Answer 3

I suggest some bottles, that they all have the same remainder of a specific number as the below figure.

may help you to visuallize.