Go backward to Quotient and Remainder Go up to Top Go forward to Exponentiation

Properties of Quotient and Remainder

• Quotient and remainder are not defined for every x and y:

  (x div 0) and (x mod 0) are undefined for every x.

• If quotient respectively remainder exist, they are unique.

  forall x, y, q0, q1, r0 < y, r1 < y:

  (x=(q0*y)+r0 /\  x=(q1*y)+r1) => (q0 = q1 /\  r0 = r1).

• If the divisor is not null, quotient and remainder exist:

  forall x, y != 0: (exists q, r: r < y /\  x = (q*y)+r).

• We thus have the following relationship:

  forall x, y != 0: x = (x div y)*y + (x mod y).