kertaluvuista

Kertaluvut, in Finnish mathematics, are the multiples of a given positive integer. For a positive integer n, the kertaluvut of n are the numbers n, 2n, 3n, 4n, and so on—every integer that can be written as n multiplied by another positive integer. The set of kertaluvut of n is infinite and forms an arithmetic progression with common difference n.

Properties of kertaluvut include closure under addition and under multiplication by positive integers: if x is a kertaluku of n, so is kx for any positive integer k. In divisibility terms, n divides every kertaluku of n. The concept is often used to study common multiples and modular relationships, since every kertaluku of n is congruent to 0 modulo n.

Common multiples of two or more integers can be described in terms of kertaluvut. The set of common kertaluvut of a and b is precisely the set of multiples of the least common multiple, lcm(a, b). Conversely, the greatest common divisor, gcd(a, b) relates to the structure of shared divisors, while the multiples concept helps identify which numbers are divisible by a given set of integers.

Examples illustrate the idea: the kertaluvut of 4 are 4, 8, 12, 16, …; the kertaluvut of 6 are 6, 12, 18, 24, … . Understanding kertaluvut aids in solving divisibility problems, simplifying arithmetic, and framing modular equations in elementary number theory. See also multiples, lcm, gcd, and arithmetic progressions.