Divisibility

For integers a and b, a divides b when b equals a times an integer. This relation underlies factors, multiples, and much of elementary number theory.

Integer: A whole number, positive, negative, or zero. Divisibility is defined between integers, not arbitrary real numbers.
Euclidean Algorithm: A procedure for finding the greatest common divisor by repeated division with remainder. It reduces common-divisor calculations to a sequence of divisibility checks.
Least Common Multiple: The least positive integer that is a multiple of each of two or more integers. It identifies the smallest shared multiple using the same divisibility relation.
Rational Number: A number expressible as the ratio of two integers with nonzero denominator. Unlike integer divisibility, rational ratios between nonzero integers always exist.
Factor: An integer that divides another integer without a remainder. A divisor is precisely a factor of the number it divides.
Division Algorithm: The theorem that expresses an integer as a divisor times a quotient plus a bounded remainder. It formalizes the quotient-and-remainder test for divisibility.
Fundamental Theorem of Arithmetic: The theorem that every integer greater than one has a unique prime factorization, up to order. Its uniqueness makes divisibility reducible to comparing prime factors.
Irrational Number: A real number that cannot be expressed as a ratio of integers. It lies outside the integer domain where the usual divisibility relation is defined.
Multiple: A number obtained by multiplying a given number by an integer. Every integer divisible by a number is one of its multiples.
Prime Factorization: The expression of an integer greater than one as a product of primes. Comparing prime exponents gives a direct test for whether one positive integer divides another.

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