Edit: I don't think I've expressed myself very well. This is another way of putting my question.
We can say two integers, a and b, are congruent mod m (where m is a natural number) if both numbers divided by m produce the same remainder. In other words, m must evenly divide their difference, a - b
Now, why are the two statements equivalent? (Why can we say "in other words"?)
How to prove
$m\mid (a-b) \iff$ $a/m$ and $b/m$ have the same remainder
*** Original question below.
I understand that one definition of integer congruence is:
For a positive integer n, two numbers a and b are said to be congruent modulo n, if their difference a − b is an integer multiple of n
I also understand that the modulus operation (for mod n) can be seen as the remainder on division by n.
What I can't see is how these two concepts relate. Could someone please explain (perhaps using a number line) why the first definition holds?