factorshave

Factorshave is a binary relation on positive integers defined as follows: a factorshave b if every prime divisor of a also divides b. Equivalently, the set of distinct prime factors of a is a subset of the set of distinct prime factors of b. The relation depends only on the prime support (the radical) of the numbers, not on the exponents of the primes.

Basic properties include reflexivity and transitivity: every number factorshave itself, since its prime set is a subset of itself, and if a factorshave b and b factorshave c, then a factorshave c. It is not antisymmetric, because two numbers with identical prime supports may each factorshave the other; for example, 12 and 18 both have prime factors {2, 3}, so 12 factorshave 18 and 18 factorshave 12.

Relation to divisibility: whenever a divides b, a factorshave b, since every prime divisor of a also divides b. However, the converse is not generally true; a factorshave b does not guarantee that a divides b, because exponents of common primes may differ.

Examples: 8 factorshave 6 because {2} is a subset of {2,3}. Conversely, 6 does not factorhave 8 since {2,3} is not a subset of {2}. Numbers with the same prime support, such as 12 and 18, factorhave each other.

Computation and related concepts: to test factorshave, factor each number and compare the sets of its prime divisors. A closely related notion uses rad(n), the product of distinct prime factors of n; a factorshave b if and only if rad(a) divides rad(b).

See also: divisibility, radical of an integer, prime factorization.