Quotient Remainder Theorem


The quotient remainder theorem, also often called the division algorithm, deals with Euclidean division: the division of one integer by another resulting in a quotient and a remainder. The theorem is a way of restating the result of long division. It states that:

Given any integer a and an integer b \neq 0, there exists two unique integers q and r such that:

a = bq + r


0 \leq r < |b|.

a is the dividend, b is the divisor, q is the quotient, and r is the remainder.

The theorem takes this form because in Euclidean division a, b, q, and r must all be integers. Ignoring this constraint and rearranging the equation by dividing both sides by b yields an equation that may better illustrate the variables and their roles:

\frac{a}{b} = q + \frac{r}{b}

The result of division is undefined when the divisor b = 0.

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