It can be proved that there exists a unique integer quotient `` q '' and a unique floating-point remainder `` r '' such that `` a '' = `` qd '' + `` r '' with 0 ≤ `` r '' < | `` d '' | .
It can be proved that there exists a unique integer quotient `` q '' and a unique real remainder `` r '' such that `` a '' = `` qd '' + `` r '' with 0 ≤ `` r '' < | `` d '' | .