This is one of the first lines in one of my lecture notes, where they write:
$x = x' \pmod N$ if and only if $N$ divides $x - x'$
I've taken a discrete maths course a while ago but this doesn't make much sense to me. I can't remember seeing $x'$ anywhere. If anyone can dumb this down or link a youtube video I'd appreciate it.
On
That's a definition.
Perhaps using other letters helps:
Given integers, $a,b,n$, we say that $a \equiv b \bmod n$ when $n$ divides $a − b$.
The notation is suggestive of equality and it does share many of its properties, especially with respect to arithmetic. This suggestive notation is due to Gauss.
See Wikipedia for examples and further explanations.
On
It's a lot less mystical than you might think. You have one number called $x$, and you have another, a priori completely unrelated number called $x'$ (it could've been called $y$ or $a$ or $\xi$ or something else, but this time it's called $x'$), and your notes contain the definition of what the statement "$x\equiv x'\mod N$" means.
On
If its the notation you xan replace $x^{\prime}$ with $y$.
Some people like to think of modulus as a "remainder function, and if that might be the way you originally learned it:
If $x$ and $y$ have the same remainder when divided by $N$, then $Nk+r=x$ and $Nl+r=y$ for integers $k,l$. So when you subtract them, $x-y=Nk+r-Nl-r=N(k-l)$, so the result will divide n. But to have the same remainder means they are congruent mod $N$
You can easily prove that intuitively "having the same remainder" and this definition are equivalent.
This is the definition of mod. $x$ and $x^{\prime}$ are arbitrary. For example, $$ 7\equiv3\mod4. $$ In this case, $x=7$ and $x^{\prime}=3$. You can check that the above is true (with respect to the definition you are given) by noting that $x-x^{\prime}=7-3=4$ is divisible by $4$.