Nondivisibility

Nondivisibility describes the property of an integer a not dividing another integer b. Under the standard definition, a divides b if there exists an integer k with b = a k. Nondivisibility is the negation of this statement: there is no such integer k. In many treatments, divisors are taken to be nonzero, in which case 0 is not considered a divisor of any nonzero number, while every nonzero a divides 0.

Examples illustrate the concept. 4 does not divide 6, since no integer k satisfies 6 = 4 k. 5 does not divide 12. By contrast, 4 divides 8 and 7 divides 21. For negative numbers, divisibility is defined similarly: -4 divides 8 because 8 = (-4)(-2). In natural-number contexts, b is a multiple of a if b equals a, 2a, 3a, and so on; nondivisible pairs are those not in that set.

Basic properties and conventions. For a fixed nonzero a, exactly the multiples of a satisfy a | b, and among natural numbers the proportion of such b is about 1/|a|, as numbers grow large. Divisibility is transitive: if a | b and b | c, then a | c. Nondivisibility does not exhibit simple closure properties under addition or multiplication, but it is the natural negation of the divisibility relation and is used whenever one must assert that no quotient exists.

Relation to broader theory. Nondivisibility is a fundamental predicate in elementary number theory, underpinning concepts such as primes, gcds, and modular arithmetic. It also appears in combinatorial constructions and in proofs that require demonstrating the absence of a particular divisor.