Though Remainders are never negative but somtime some questions of finding the remainder become easy by using Negative Remainders. 
First of all, What is Negative Remainder:
eg. When 14 is divided by 8
Then, 14= 8×1+6
or
14=8×2-2
Therefore, We have two remainder and they are +6 and -2 where (-2) is the Negative Remainder. I hope I am able to make it clear.
Now, I make a small assumption that you know the remainder theorum and you make use of it while calculating the remainder. Ok, Now lets take an example
Q. Find the remainder when 14×15 is divided by 8
Ans. Traditional method: Find remainder after dividing 14 and 15 by 8, multiply them and find the final answer i.e
a)14/8= 8×1+6
b)15/8=8×1+7
c)find remainder after dividing 6×7 by 8 = 42/8=8×5+2
Therefore, Remainder is 2
New method,
a)14=8×2-2 (minus 2 is the negative remainder)
b)15=8×2-1 ( minus 1 is the negative remainder)
Final Remainder is = (-2)x(-1) divided by 8 =2
Therefore Remainder is 2.
Though both the method looks same in calculation level but when you some difficult problem then this negative remainder method will help a lot and will save some time.s
Remember: If at the end remainder is negative then substract that number from the divisor for eg. If the answer was (-3) then remainder would be = 8-3=5.
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