I do not see where this would be of any use.

On the other hand, if you want a quotient that has some meaningful relationship with the ratio between the dividend and the divisor, there are other more sensible definitions of the integer division than the one used in modern programming languages.

You can have either a result that is a floating point number even for integer dividend and divisor, like in Algol, or you can define the division to yield the quotient rounded to even (i.e. with a remainder that does not exceed half of the divisor).

In both cases 10/3 and -10/-3 would yield the same result and I can imagine cases when that would be useful.

For the current definition of the integer division, I do not care whether 10/3 and -10/-3 yield the same result. It does not simplify any algorithm that I am aware of, while having a remainder of a known sign simplifies some problems by eliminating some tests for sign.