How can I solve for mod to get a value? I want to solve the following part involving mod:
1 = -5(19) (mod 96)

Apparently, this mod in brackets (mod 96) here is different from the mod that I know e.g. its not the remainder value that you get by dividing.
What of kind of mod is it and how can I solve it step by step to get the result which is 77?
Update:
Okay, so I am trying to find what's 5 inverse mod 96.
By following euclidean algorithm approach here's what I am doing:
Step1: Find GCD of 5 and 96
96 = 5(19) + 1

which becomes 1 = 96-5(19) when expresses in 1 term
Step 2: Take mod (96) both sides
So I will have left:
1 = -5(19) (mod 96)

That's from where I need to solve it to get 5 inverse (mod 96).   

$1 \equiv -5(19) \mod 96$ is equivalent to $1 \equiv -95 \mod 96$ by multiplying out the brackets. 
For your question it's equivalent to going round a clock with $96$ hours on it - each time you reach $96$th hour it goes back to $0$ and starts again. 
This can also be extended to negative numbers, and adding multiples of $96$. Hence:
$$1 \equiv -5(19) \mod 96 \equiv 1 \mod 96$$
