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There are several ways, choose the one you prefer.
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Don't forget the braces: try a\equiv b \pmod 11 or a\equiv b \pmod pq and see why.
The last one is marked “wrong”, because the usage is improper: \bmod should be used for the “modulo” binary operation (the one that is often denoted by % in computing).
Use pmod n. It is known to work.
- I am a human being.
- You are a human being.
Therefore, I am you: right ? Well, as it turns out, the answer is no. It simply means that we belong to the same class. Likewise, 
, but 
.
I prefer to use the congruence exclusively (I view the remainder operation as a source of headache even though I realize that it is a necessary evil in computing).
But I use both equal (
) and congruent to (
) signs together when processing a lengthy calculation in modular arithmetic. My freshman algebra students quickly catch on with my calculations like
$$
12^{3004}\equiv5^{3004}=5^{3000}\cdot 5^4=(5^6)^{500}\cdot 5^4\equiv 1^{500}\cdot5^4=25^2\equiv4^2=16\equiv2\pmod7.
$
=$ when there is an equality of integers between the steps, and 
when I mean a congruence. The power of laws obeyed by congruences is apparent. Checking/following the progress of the calculation is easier this way. Of course, using 
all the way is correct also. The 
is there as a reminder that in this step we do something that only results in a congruence.
As the students become acquainted with the language of residue class rings, I gradually stop making the distinction between 
and 
as well as, clarity of context permitting, the distinction between 
and 
.
Thinking about what that would look like when done by somebody who is only familiar with binary mod makes me shudder.
Hello everyone,
In modular arithmetic, if we know the remainder r when dividing a by m, we write it as:
a ≡ r mod m
As I understand it, r is the result of the operation a mod m.
However, in other formulas—like in RSA encryption—we often see something like:
y ≡ x^(e) mod m
This means that y is the result of the operation x^(e) mod n.
So to me, it would feel more intuitive to write:
x^(e) ≡ y mod n
since x^(e) mod n = y, and the expression being reduced appears on the left-hand side.
The way the modular expression is written can be a little confusing at first, but both forms describe the same relationship.