How can I find a mod with negative number? I know how to solve mod using division i.e.
$$11 \mod 7 = 4$$
For this I did a simple division and took its remainder:
i.e.
$$11 = 7 \cdot 1 + 4$$
Where $11$ was dividend, $7$ divisor, $1$ quotient and $4$ was remainder.
But I have a problem with:
$$-11 \mod 7 = 3$$
How come it is $3$? I cannot figure this out using division but if it is possible I would like to.
 A: It's $3$ because $-11 = 7(-2) + 3$.
Another way to see this is to take $-11$ and keep adding $7$ to it until you get a positive number.  This works because, if you're working modulo $7$, then adding $7$ is the same as not changing the number (modulo $7$).  So:
$-11 + 7 \equiv -11 \pmod 7$, and $-11 + 7 = -4$.  Therefore $-4 \equiv -11 \pmod 7$.  Well, we're still negative.  Let's do it again:
$-4 + 7 \equiv -11 \pmod 7$, and $-4 + 7 = 3$.  Therefore, $3 \equiv -11 \pmod 7$.
Or, equivalently, $-11 \equiv 3 \pmod 7$.

How do we know to use $-2$?  Let's recall how it works with positives first.
If you want to evaluate $31 \pmod 7$, you first recognize that $31 = 28 + 3 = 7 \cdot 4 + 3$.  Therefore $31 \equiv 3 \pmod 7$.  What did we do here?  We found the largest multiple of $7$ that's less than or equal to $31$.
Alternatively, with division, you can evaluate $31/7 \approx 4.429$.  The largest integer less than or equal to this is $4$.  Therefore $31 = 7 \cdot 4 + \text{some number}$, where your goal is to determine what $\text{some number}$ is.
This same exact process applies for negative numbers.
If you want to evaluate $-11 \pmod 7$, you need the largest multiple of $7$ that's less than or equal to $-11$.  This is $-14$.  And $-14 + 3 = -11$, therefore your answer is $3$.
Alternatively, with division, you can evaluate $-11/7 \approx -1.571$.  The largest integer less than or equal to this is $-2$.  Therefore $-11 = 7 \cdot (-2) + \text{some number}$, where your goal is to determine what $\text{some number}$ is.
A: I regard the negative remainder as an overshoot on the division operation. Since by definition the remainder needs to be positive, then another operation is required to get the "true remainder".
e.g. 43=-2 (mod5)
You can express as above 43 = 8x 5 + 3, or 43 = 9x5 - 2. In this last expression you are "short" 2 to complete the module 5, but you are "over 3" to the previous complete module 40. Hence the positive remainder is 5-2 =3 (i.e. Module plus the negative remainder).
Operationally would be to use the standard division, but note that the remainder is negative, then you need to do the last operation to get the positive remainder.
In your example:
-11 mod 7 = 3
-11/7 = - 1 4/7 , My reaction is regard 4 as the remainder. However note that it is really -4. Hence you need to do the last operation: 7-4 = 3
I apologize for the lack of rigor on my use of definitions, but it's just a way to figure this logically.
