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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 ...
In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient. Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 / 3 leaves a remainder of 1, as 10 is not a multiple of 3.
Find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once. In this case, this is simply the first digit, 5. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, so the digit 1 is put above the 5 to start constructing the quotient.
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.
In this case, s is called the least absolute remainder. [3] As with the quotient and remainder, k and s are uniquely determined, except in the case where d = 2n and s = ± n. For this exception, we have: a = k⋅d + n = (k + 1)d − n. A unique remainder can be obtained in this case by some convention—such as always taking the positive value ...
The quotient and remainder can then be determined as follows: Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of x, which in this case is x). Place the result above the bar (x 3 ÷ x = x 2). ) ) + ¯ Multiply the divisor by the result just obtained (the first term of the eventual ...
Next one repeats step 2, using the small digit concatenated with the next digit of the dividend to form a new partial dividend (15). Dividing the new partial dividend by the divisor (4), one writes the result as before — the quotient above the next digit of the dividend, and the remainder as a small digit to the upper right.
If the last group is a remainder smaller than the divisor, it can be thought of as forming an additional smaller group. For example, if 45 eggs are to be put into 12-egg cartons, then after the first 3 cartons have been filled there are 9 eggs remaining, which only partially fill the 4th carton.
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