I am learning modular arithmetic and trying to figure out, how to find remainder where denominator is greater than numerator?

For example:

i) $2 \bmod 5 =$ ?

I tried to solve this but I got 0 as remainder whereas in calculators it is $2$ . I was solving it with regular math operators like adding 0 and value after points.

ii) $-2 \bmod 5$ = ?

Also I wanted to know, how to handle negative number in modular arithmetic?

  • $\begingroup$ $2\pmod{\!5}\ $ is meaningless. It should be $\,2\bmod 5\,$ if it denotes the remainder when $2$ is divided by $5$. It is essential to understand the difference between $\!\bmod\!$ as a binary operation (remainder) vs. ternary congruence relation, e.g. see this answer, and this one by Arturo Magidin. $\endgroup$ – Bill Dubuque Sep 25 '17 at 13:28
  • $\begingroup$ @BillDubuque I wrote 2 mod 5 but it was changed to 2 (mod 5) by N. F.Taussig. Please check edit history. $\endgroup$ – Alena Sep 28 '17 at 5:00

Remember, by definition, the remainder when dividing $m/n$ is such a number $r$ such that

  1. $0\leq r<n$
  2. There exists some $k$ such that $k\cdot n + r = m$

by that definition, the remainder when dividing $2$ by $5$ is $2$, because $$0\cdot 5 + 2 = 2$$

As far as modular arithmetic is concerned, remember that $$x\equiv y\mod n$$ if and only if $n|x-y$. As a consequence, it is always true that $$x+k\cdot n\equiv x\mod n$$ for any integer $k$. In your case, that means that $$-3\equiv -3+1\cdot 5\mod 5$$

  • $\begingroup$ thanks for helpful answer, how 0.5 comes? when we divide after adding .0 it becomes 20 and 20-20 is equal to 0. Looks like one of my step is wrong and wanted to correct it $\endgroup$ – Alena Sep 25 '17 at 12:52
  • $\begingroup$ @Alena $0.5$ is a rational number and has nothing to do with this question. But by definition, $0.5$ is equal to $\frac 5{10}$. $\endgroup$ – 5xum Sep 25 '17 at 12:54
  • $\begingroup$ @5xum In some countries, the product of $a$ and $b$ is written $a.b$. $\endgroup$ – N. F. Taussig Sep 25 '17 at 12:59
  • $\begingroup$ @Alena If by $0.5$ you mean $0\cdot 5$, then the $0$ comes from the fact that $0$ is the number $k$ for which $k\cdot 5 + 2 = 2$. You can calculate the number by integer division - $2$, divided by $5$, is equal to $0$. $\endgroup$ – 5xum Sep 25 '17 at 13:00
  • $\begingroup$ Please don't abuse notation in a way that heightens the confusion between $\!\bmod\!$ the relation vs. operation. Correct is $\,x\equiv y\pmod{\!n}.\,$ What you wrote is easily confused with $\, x = y\bmod n,\,$ i.e. $\,x = (y\bmod n)\,$ by beginners. $\endgroup$ – Bill Dubuque Sep 25 '17 at 13:18

Here is a realisation that helped me out back when I was learning modular arithmetic: Don't think of the symbol $\!\!\!\pmod k$ as an operation, but rather as part of the relation $\equiv$. For instance, $$ 7\equiv 2\pmod 5 $$ doesn't mean that if you take $7$ and apply the "modulo five" operation to it, you get $2$ (or the other way around). It rather means that $7$ and $2$ are in some sense "five-similar", i.e. they are both equally far away from any multiples of $5$. Think $7=_5 2$, if that makes any sense to you.

That way, you can also use, for instance $$ 1974 \equiv -1\pmod 5 $$ which might be helpful since $-1$ is often much easier to handle than $4$ in a lot of arithmetic, like if you were told to calculate $1974^{37}$ modulo $5$.

PS. The phrase "five-similar" is just something I came up with right now, and is not in any way conventional terminology. Unless you explain it, there is real risk that no one will understand it.


We say that $a \equiv b \pmod{n}$ if $a$ and $b$ have the same remainder when divided by $n$. Equivalently, $a \equiv b \pmod{n}$ if $n \mid a - b$.

By the Division Algorithm, given $m, d \in \mathbb{Z}$, with $d \neq 0$, there exist $q, r \in \mathbb{Z}$ such that $m = qd + r$ and $0 \leq r < |d|$. The number $q$ is called the quotient; the number $r$ is called the remainder.

Since $2 = 0 \cdot 5 + 2$, $2$ has quotient $0$ and remainder $2$ when divided by $5$. Hence, $2 \equiv 2 \pmod{5}$.

Since $-2 = -1 \cdot 5 + 3$, $-2$ has quotient $-1$ and remainder $3$ when divided by $5$. Hence, $-2 \equiv 3 \pmod{5}$.

  • $\begingroup$ could you please explain? that doesn't make any sense to me. $\endgroup$ – Alena Sep 25 '17 at 11:47
  • $\begingroup$ I elaborated. Let me know if it still does not make sense to you. $\endgroup$ – N. F. Taussig Sep 25 '17 at 12:58

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