Congruence modulo

Congruence Modulo

You may see an expression like:
$A \equiv B (\text{mod } C)$
This says that $A$ is congruent to $B$ modulo $C$.
We will discuss the meaning of congruence modulo by performing a thought experiment with the regular modulo operator.
Let's imagine we were calculating mod 5 for all of the integers:
Suppose we labelled 5 slices 0, 1, 2, 3, 4. Then, for each of the integers, we put it into a slice that matched the value of the integer mod 5.
Think of these slices as buckets, which hold a set of numbers. For example, 26 would go in the slice labelled 1, because $26 \text{ mod } 5 = \bf{1}$.
Above is a figure that shows some integers that we would find in each of the slices.
It would be useful to have a way of expressing that numbers belonged in the same slice. (Notice 26 is in the same slice as 1, 6, 11, 16, 21 in above example).
A common way of expressing that two values are in the same slice, is to say they are in the same equivalence class.
The way we express this mathematically for mod C is: $A \equiv B \ (\text{mod } C)$
The above expression is pronounced $A$ is congruent to $B$ modulo $C$.
Examining the expression closer:
1. $\equiv$ is the symbol for congruence, which means the values $A$ and $B$ are in the same equivalence class.
2. $(\text{mod } C)$ tells us what operation we applied to $A$ and $B$.
3. when we have both of these, we call “$\equiv$congruence modulo $C$.
e.g. $26 \equiv 11\ (\text{mod }5)$
$26 \text{ mod } 5 = 1$ so it is in the equivalence class for 1,
$11 \text{ mod } 5 = 1$ so it is in the equivalence class for 1, as well.
Note, that this is different from $A \text{ mod } C$: $26 \neq 11 \text { mod } 5$.

Insights into Congruence Modulo

We can gain some further insight behind what congruence modulo means by performing the same thought experiment using a positive integer $C$.
First, we would label $C$ slices $0, 1, 2, \ldots, C - 2, C - 1$.
Then, for each of the integers, we would put it into a slice that matched the value of the integer $\text{mod } C$.
Below is a figure that shows some representative values that we would find in each of the slices.
If we looked at the bucket labelled 0 we would find:
$\ldots, -3C, -2C, -C, 0, C, 2C, 3C, \ldots$
If we looked at the bucket labelled 1 we would find:
$\ldots, 1-3C, 1-2C, 1-C, 1, 1+C, 1+2C, 1+3C, \ldots$
If we looked at the bucket labelled 2 we would find:
$\ldots, 2-3C, 2-2C, 2-C, 2, 2+C, 2+2C, 2+3C, \ldots$
If we looked at the bucket for $C - 1$ we would find:
$\ldots, -2C-1, -C-1, -1, C-1, 2C - 1, 3C-1, 4C - 1 \ldots$
From this experiment we can make a key observation:
The values in each of the slices are equal to the label on the slice plus or minus some multiple of $\bf{C}$.
This means the difference between any two values in a slice is some multiple of $\bf{C}$.
This observation can help us understand equivalent statements and equivalence classes next.