What makes two numbers congruent?

Answer by Alan Bustany:

Two whole numbers [math]a[/math] and [math]b[/math] are said to be congruent modulo a third natural number [math]m[/math] if and only if [math]m[/math] divides the difference between [math]a[/math] and [math]b[/math]. Equivalently the remainders when the numbers are divided by [math]m[/math] are equal.
The expression "[math]a[/math] is congruent (or equivalent) to [math]b[/math] modulo [math]m[/math]" is written in symbols as follows:
     [math]a\equiv b\mod m[/math]
The usual representative for the equivalence class of a given number modulo [math]m[/math] is the natural number in the range [math][0,m)[/math] although it is often useful to use [math]-1[/math] rather than [math]m-1[/math] because of the properties of addition and multiplication modulo [math]m[/math].
Note that two numbers are congruent only modulo a third number. For example
     [math]7\equiv5\equiv1\mod2[/math]
but
     [math]7\equiv1\mod3[/math]
whereas
     [math]5\equiv2\mod3[/math]
So it does not make sense to say simply "7 is congruent to 5" without adding the modulo.

What makes two numbers congruent?

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