In arithmetic, shortdivision is a division algorithm which breaks down a division problem into a series of easy steps. It is an abbreviated form of long division — whereby the products are omitted and the partial remainders are notated as superscripts. As a result, a short division tableau is always more notationally efficient than its long division counterpart — though sometimes at the expense of relying on mental arithmetic, which could limit the size of the divisor. For most people, small integer divisors up to 12 are handled using memorised multiplication tables, although the procedure could also be adapted to the larger divisors as well. As in all division problems, a number called the dividend is divided by another, called the divisor. The answer to the problem would be the quotient, and in the case of Euclidean division, the remainder would be included as well. Using short division, one can solve a division problem with a very large dividend by following a series of easy steps.
Tableau
Short division does not use the slash or division sign symbols. Instead, it displays the dividend, divisor, and quotient in a tableau. An example is shown below, representing the division of 500 by 4. The quotient is 125. Alternatively, the bar may be placed below the number, which means the sum proceeds down the page. This is in distinction to long division, where the space under the dividend is required for workings:
Example
The procedure involves several steps. As an example, consider 950 divided by 4: Using the alternative layout the final workings would be: As usual, similar steps can also be used to handle the cases with a decimal dividend, or the cases where the divisor involves multiple digits.
A common requirement is to reduce a number to its prime factors. This is used particularly in working with vulgar fractions. The dividend is successively divided by prime numbers, repeating where possible: So 950 = 2 x 5² x 19
Modulo division
When one is interested only in the remainder of the division, this procedure ignores the quotient and tallies only the remainders. It can be used for manual modulo calculation or as a test for even divisibility. The quotient digits are not written down. For example, what is the remainder of 16762109 divided by 7? The remainder is zero, so 16762109 is exactly divisible by 7.
