Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In arithmetic, when the result of the division of two integers cannot be expressed with an integer quotient, the remainder is the amount "left over." If a and d are natural numbers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder. The division algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder.