divisible

Divisible is a term used to describe a relationship between integers called divisibility. An integer a is said to divide another integer b if there exists an integer k with b = a times k. If this holds, we denote it as a | b and say "a divides b." The numbers a and b are then called the divisor and the multiple of each other.

Examples: 3 | 12 because 12 = 3 times 4. Also, -3 | 12 since 12 = (-3) times (-4). On the other hand, 5 does not divide 14 because no integer k satisfies 14 = 5 times k. By convention, divisibility is often discussed with positive divisors, though negative divisors are also valid.

Special cases: every nonzero a divides 0, since 0 = a times 0. The relation is defined for a ≠ 0 in the standard formulation, because 0 is not used as a divisor of nonzero numbers.

Related concepts: the set of divisors of a nonzero integer n is finite; the count is denoted d(n) or τ(n). A divisor relationship is transitive: if a | b and b | c, then a | c. The quotient b / a is an integer whenever a | b. Divisibility underpins concepts such as greatest common divisor (gcd) and least common multiple (lcm), and is central to prime factorization, where primes are numbers with exactly two positive divisors: 1 and themselves.