But we ain't talking about higher mathematics here, that's plain algebra in the old 10-based numerical system, the dear old mathematics we studied in the early years...

No, actually we're not talking about that. We're talking about programming, so we're talking about what computers do. And what most of them do nowadays is implement a single instruction that does a truncating divide and gives you a quotient and remainder from that operation. Some extended-precision arithmetic packages (like GMP) give you both round-towards-zero and round-towards-negative-infinity (in other words, positive remainder) division routines, but that's as maybe.