. It can also written by simple juxtaposition: ab. In computer programming languages and software packages (in which one can only use characters normally found on a keyboard), it is often written with an asterisk: a * b. Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and applying repeated addition, or on employing tables or slide rules, thereby mapping multiplication to addition and vice versa. These methods are outdated and are gradually replaced by mobile devices. Computers use diverse sophisticated and highly optimized algorithms, to implement multiplication and division for the various number formats supported in their system. Division Division, denoted by the symbols or , is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Under common rules, dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than or equal to 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend. Division is neither commutative nor associative. So as explained in , the construction of the division in modern algebra is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced in . Hence division is the multiplication of the dividend with the reciprocal of the divisor as factors, that is, Within the natural numbers, there is also a different but related notion called Euclidean division, which outputs two numbers after "dividing" a natural (numerator) by a natural (denominator): first a natural (quotient), and second a natural (remainder) such that and In some contexts, including computer programming and advanced arithmetic, division is extended with another output for the remainder. This is often treated as a separate operation, the Modulo operation, denoted by the symbol or the word , though sometimes a second output for one "divmod" operation. In either case, Modular arithmetic has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc.) correspond with different implementations of modulus. Fundamental theorem of arithmetic The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number