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 congruentonlymodulo 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 doesnotmake sense to say simply "7 is congruent to 5" without adding the modulo.

What makes two numbers congruent?

Advertisements