Clarifying meaning of "mod" I'm having trouble understanding exactly how $mod$ works. I know it's remainder, but I guess my problem is in how to write it out.

For example, $\frac{1}{3} = 0$ remainder $1$. 

Would I write $1 = 0(mod 3)$, $1 = 1 (mod 3)$...?

Edit: So, the base goes in the (mod) brackets, the number being divided is on the left, and the remainder is on the right?
Though the question is already “answered”, it’s worth noting there are two uses of $\bmod$:


$a\bmod b$. This is a number, the remainder when dividing $a$ by $b$.
$a\equiv a'\pmod b$. This is a statement “$a$ is congruent to $a'$ modulo $b$”, that is, $a-a'$ is a multiple of $b$. Here, $a'$ can, but doesn’t need to be equal to $a\bmod b$. For example, $3\equiv 7\pmod2$. The number $b$ is called modulus in this case.

Sometimes, $a\pmod b$ is used. It means a residue class modulo $b$ which contains $a$, that is, a set of all $a'$ such that $a\equiv a'\pmod b$.


By the way, in the latter case, one can also write many congruences together, e. g. $28\equiv 8\equiv -12\equiv 578\pmod{10}$, like it’s done with $=$ or ordering relations.

You can also speak of a zero modulus: $a\equiv a'\pmod0$ means that $a-a'$ is a multiple of $0$—that is, $a = a'$. On the other hand, typically $a\bmod0$ is treated as undefined.
Based of its definition, Modulo operation is to find the "REMAINDER AFTER DIVISION"
Since it's target is the remainder, "You mod something" = "the remainder".

You mod something is between something which in turns produce that remainder. Obviously its division. Now who divided by who gets that remainder? Those in the mod would by definition be equivalent to what we can call the BASE. 

How would you figure out for the rest? :D