Remainder when $-1$ is divided by $2$ The  two possible ways to find the reminder, 

$-1 = -1 \times 2+1$

$-1 = - 0 \times 2-1$

From the above calculation, I have found  different quotients: $+1$ and $-1$. If I am asked to  tell  that if the reminder is greater than $0$ then how can I give the answer?
Remainders are usually defined to be positive. That is, the division algorithm says that for any integers $p$ and $q$, with $q\neq 0$, there exists integers $b$ and $r$ such that

$$p=bq+r$$

where $0\le r &lt; q$.

The $0\le r&lt;q$ is the key part here. As a matter of convenience, we often conform to this by writing, for example, $-1=(-1)(2)+1$, and the remainder is $1$.
You have found something along the lines of modular arithmetic.

Imagine a clock with twelve hours.  On this clock, 9 o'clock = 21 o'clock.

In math form this would be $9=21(\mod{12})$

Imagine a different clock with only 2 hours.

We would then have $$a=2n+a(\mod2)$$The $2n$ would disappear and make no difference because it simply represents an addition of $2$ $n$-times, which disappear in modular arithmetic.