Podzielny

Podzielny is a Polish mathematical term used to describe numbers that can be divided by another number without a remainder. In precise terms, a number a is podzielny przez b if there exists an integer k such that a = b · k. Equivalently, the remainder when a is divided by b is zero. The relation is typically defined for integers; for positive integers it induces a partial order known as divisibility.

The set of integers that are podzielny przez a given number b consists of all multiples of b: ..., -2b, -b, 0, b, 2b, ... . For example, 12 is podzielny przez 3 and by 4, while 7 is not podzielny przez 2. By convention, 0 is podzielny przez any nonzero integer, since 0 = b · 0.

Basic properties include reflexivity (any number is podzielny przez itself) and transitivity (if a is podzielny by b and b is podzielny by c, then a is podzielny by c). Among positive integers, the divisibility relation defines a partial order. The concept underpins the notion of greatest common divisor (Największy wspólny dzielnik, NWD) and prime factorization, where primes are numbers whose only positive divisors are 1 and themselves.

Applications of podzielność appear across number theory and arithmetic, including simplifying fractions, checking divisibility in algorithms, and studying modular arithmetic. The related noun is podzielność, and common terminology includes dzielnik (divisor) and dzielenie (division).