Short division

Short division is a pencil and paper method of dividing one number by another in which the working is compressed into a single compact tableau instead of being written out in full. It is an abbreviated form of long division: the products that long division writes beneath the dividend are left out, and each partial remainder is carried forward by writing it beside the next digit of the dividend. The result is a tableau shorter than its long division counterpart, though the saving depends on the person holding the intermediate multiplications in their head, which in practice limits the method to small divisors, commonly those up to twelve that memorised multiplication tables cover, although it can be adapted to larger ones. Short division copes with arbitrarily large dividends and can be continued past a decimal point to give a decimal quotient. Two variations are described alongside it: reducing a number to its prime factors by dividing in turn by successive primes, and a modulo form that keeps only the remainders and so serves as a divisibility test. The remainder-tracking form can be described as a deterministic finite automaton with one state for each possible remainder, which shows that the numbers divisible by a given divisor form a regular language.