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The number q is called the quotient, while r is called the remainder. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see division algorithm.) The remainder, as defined above, is called the least positive remainder or simply the remainder. [2]
The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction). This remainder carries forward when the process is repeated on the following digit of the dividend (notated as 'bringing down' the next digit to the remainder).
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Animation showing the use of synthetic division to find the quotient of + + + by .Note that there is no term in , so the fourth column from the right contains a zero.. In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division.
Divide the highest term of the remainder by the highest term of the divisor (x 2 ÷ x = x). Place the result (+x) below the bar. x 2 has been divided leaving no remainder, and can therefore be marked as used. The result x is then multiplied by the second term in the divisor −3 = −3x. Determine the partial remainder by subtracting 0x − ...
The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...
The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique. [20] In Euclid's original version of the algorithm, the quotient and remainder are found by repeated subtraction; that is, r k−1 is subtracted from r k−2 repeatedly until the remainder r k is smaller ...
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder ...