modular arithmetic - How to find remainder of denominator is greater than numerator?

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?

Answer

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

1. $0\leq r
   
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

$$0\cdot 5 + 2 = 2$$

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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 $$-3\equiv -3+1\cdot 5\mod 5$$