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If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.}
In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.
A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only list positive divisors.
Every common divisor of a and b is a divisor of gcd(a, b). gcd(a, b), where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form d = a⋅p + b⋅q, where p and q are integers. This expression is called Bézout's identity.
In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique. Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b ...
Add twice the last digit to the rest. (Works because (10a + b) × 2 − 19a = a + 2b; since 19 is a prime and 2 is coprime with 19, a + 2b is divisible by 19 if and only if 10a + b is.) 437: 43 + 7 × 2 = 57. Add 4 times the last two digits to the rest. (Works because 399 is divisible by 19.) 6,935: 69 + 35 × 4 = 209. 20
Given elements a and b of R, one says that a divides b, or that a is a divisor of b, or that b is a multiple of a, if there exists an element x in R such that ax = b. The units of R are the elements that divide 1; these are precisely the invertible elements in R. Units divide all other elements.