Congruence ≡
Numbers are congruent when they have the same remainder after being divided by a chosen whole number (called the modulus).
Example: 2 and 14 with a modulus of 12
212 = 0 with a remainder of 2
1412 = 1 with a remainder of 2
They have the same remainder,
so 2 and 14 are congruent (modulus 12)
so 2 and 14 are congruent (modulus 12)
So they are congruent when, after spinning around any number of times, they land together.
Example: 17 and 134 mod 9
179 = 1 with a remainder of 8
1349 = 14 with a remainder of 8
They have the same remainder,
so 17 and 134 are congruent (modulus 9)
so 17 and 134 are congruent (modulus 9)
The symbol is ≡
17 ≡ 134 mod 9
"17 is congruent to 134 using a modulus of 9"
Play with it here:
images/cong-anim.js
Another Way of Looking at It
We can also subtract the numbers and see if the result is a multiple of the modulus.
Example: 2 and 14 mod 12 (again)
14 − 2 = 12
12 / 12 = 1 exactly
Example: 17 and 134 mod 9 (again)
134 − 17 = 117
117 / 9 = 13 exactly