Should % ever "overflow"?

[Smoke Adams via Digitalmars-d]

On Friday, 24 June 2016 at 20:43:38 UTC, deadalnix wrote:
> On Friday, 24 June 2016 at 20:33:45 UTC, Andrei Alexandrescu 
> wrote:
>> In a checked environment, division may "overflow", e.g. -6 / 
>> 2u must be typed as uint but is not representable properly one.
>>
>> How about remainder? I suppose one can make the argument that 
>> remainder should never overflow (save for rhs == 0), because 
>> it could be defined with either a positive or negative 
>> denominator (which is not part of the result).
>>
>> What's the most reasonable behavior?
>>
>>
>> Andrei
>
> Most reasonable is
>
> numerator = quotient * divisor + remainder
>
> Which means it can be negative.

This proves nothing.

Wiki:

For a ***positive integer n***, two integers a and b are said to 
be congruent modulo n, written:

     a ≡ b ( mod n ) , {\displaystyle a\equiv b{\pmod {n}},\,} 
a\equiv b{\pmod {n}},\,

if their difference a − b is an integer multiple of n (or n 
divides a − b). The number n is called the modulus of the 
congruence.


For any "negative" remainder, all one has to do is subtract one 
from the divisor:

quotient*(divisor - 1 + 1) + remainder
= quotient*new_divisor + pos_remainder

where new_divisor = divisor - 1, pos_remainder = quotient + 
remainder

There are problems with allowing the remainder to be negative and 
it is generally best to restrict it to positive values.

e.g., 129 = 2*52 + 25 OR 3*52 - 27.

In the second case, we have 129/52i = 3? Basically by restricting 
to positive integers, we have a more natural interpretation and 
relationship to floor and ceil functions and we don't have to 
worry about it being negative(think of using in in an array 
indexing scheme),