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:

\equiv
is the symbol for congruence, which means the valuesA
andB
are in the same equivalence class. 
(\text{mod } C)
tells us what operation we applied toA
andB
. 
when we have both of these, we call “
\equiv
” congruence moduloC
.
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, 13C, 12C, 1C, 1, 1+C, 1+2C, 1+3C, \ldots
If we looked at the bucket labelled 2 we would find:
\ldots, 23C, 22C, 2C, 2, 2+C, 2+2C, 2+3C, \ldots
If we looked at the bucket for C  1
we would find:
\ldots, 2C1, C1, 1, C1, 2C  1, 3C1, 4C  1 \ldots
From this experiment we can make a key observation:
The values in each of the slices are equal to the 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.