nonmultiple

Nonmultiple is a term used primarily in mathematics and computer science to describe a relationship between two integers in which one integer is not an integer multiple of the other. Formally, given two integers a and b, a is a nonmultiple of b if there exists no integer k such that a = k·b. Equivalently, the condition can be expressed as a mod b ≠ 0, where the modulus operation yields a non‑zero remainder.

The concept is frequently employed in problems involving divisibility, factorization, and modular arithmetic. In number‑theoretic contexts, determining whether a number is a nonmultiple of another can be essential for testing primality, constructing coprime pairs, and analyzing residue classes. In algorithm design, nonmultiple checks often appear in loops that skip over numbers sharing a common factor, such as the sieve of Eratosthenes for generating prime numbers.

Examples illustrate the notion: 7 is a nonmultiple of 3 because 7 mod 3 = 1; 12 is not a nonmultiple of 4 because 12 mod 4 = 0. The term may also be extended to sets: a set S contains a nonmultiple of b if at least one element of S fails to be divisible by b.

Related concepts include multiple, divisor, factor, greatest common divisor, and coprime. Though “nonmultiple” is not a standard entry in most mathematical dictionaries, it serves as a useful descriptive label in teaching and computational contexts where explicit statements about non‑divisibility are required.