Difference between modulus and remainder

Question

_{(I've come here from this question.)}

What's the difference between modulus and remainder?

^{(Bearing in mind until 5 minutes ago I thought they were the same :P )}

Answer 1

Let's say we have integers $a $ and $b$. The $remainder$ of $a$ divided by $b$ is the number $r$ such that $a = qb +r$ for some $q\in\mathbb{Z}$ (here $\mathbb{Z}$ is the set of integers), and $0\leq r <b$. In this situation, $r$ is uniquely determined. That is, there is only one value of $r$ which satisfies these requirements, and so we say that $r$ is $the$ remainder of $a$ divided by $b$.

Modulus, in this situation, refers to the number $b$. However, given the context, I think what you meant to refer to was "$a$ mod $b$", or the class of $a$ modulo $b$, which is actually an infinite set of integers. Precisely, the class of $a$ modulo $b$ is the set \begin{equation*} a+b\mathbb{Z} = \left\{a+bn\;|\;n\in\mathbb{Z}\right\}.\end{equation*} Every number in this set yields the same remainder after division by $b$. So, for example, $5$ mod $7$ is the same as $12$ mod $7$, because we have the equality of sets \begin{equation*} 5+7\mathbb{Z} = 12 + 7\mathbb{Z}. \end{equation*} Often, people say "$12 $ mod $7 = 5$," which is technically incorrect. What we should say is "the class of 12 mod 7 is equal to the class of 5 mod 7", or "5 and 12 are equivalent modulo 7". When working modulo $n$, we have a preferred set of representatives for each class modulo $n$, which is usually \begin{equation*} \{0,1,...n-1\}.\end{equation*}That means that every number is equivalent to one of these numbers modulo $n$. When we use this particular set of class representatives, they just so happen to coincide with the usual notion of remainder.

Answer 2

The modulus in a modular equation $\pmod{n}$ is the number $n$. That is exactly what "mod n" refers to.

The set of integers with the same remainder forms an equivalence class that's sometimes called "the residue modulo $n$."

For and integer $a$ and a positive integer $b$, the division algorithm says that there exists unique integers $q,r$ such that $a=bq+r$ and $0\leq r<b$.

They are related in the sense that $a\equiv a'\pmod b$ iff $a$ and $a'$ have the same remainder after division by the modulus $b$.

Answer 3

Modulus can refer to a couple of different things depending on the context. There is modulus in the sense of modular arithmetic (the base number under consideration), there is modulus of complex number (think absolute value) and there is a notion of modulus in algebraic number theory.