Factorsinto

Factorsinto is a binary relation used in arithmetic and number theory to express that one integer is a factor of another. In standard terms, a factors into b if and only if there exists an integer k such that b = a × k. Equivalently, a factors into b when a divides b, often denoted as a | b.

Formal properties include reflexivity on nonzero integers (a factors into a) and, for positive integers, the divisibility relation provides a partial order: if a factors into b and b factors into c, then a factors into c. If a divides b and b divides a for positive integers, then a equals b (antisymmetry). The relation is left- and right-cancellable in many algebraic contexts and interacts with gcd and lcm computations.

Usage and interpretation: factorsinto is commonly used when discussing divisibility, multiples, and factorization. Examples include 3 factors into 12, 5 factors into 25, while 6 does not factor into 20 because 20 is not a multiple of 6.

Computational aspects: in programming, a factors into b can be checked by testing whether b mod a equals zero, or by performing integer division to verify b = a × floor(b/a). Many languages provide a modulus operation to test divisibility directly.

Variants and terminology: a is a proper factor of b if a divides b and a ≠ b. Some contexts describe factorization or decomposition using phrasing related to “factors into,” though the underlying concept remains divisibility. Notation can vary, but the relation corresponds to the standard divisibility relation on integers. See also divisibility, factorization, gcd, lcm, and modular arithmetic.