Faktorer

Faktorer, in mathematics, are numbers or expressions that divide another number or expression exactly. For a positive integer n, a factor is a positive integer d such that n divided by d is an integer. Every positive integer has a finite set of positive factors; 1 and the number itself are always factors. For example, the positive factors of 12 are 1, 2, 3, 4, 6, and 12. Negative factors also exist, since multiplying two negatives yields a positive result.

Factorization is the process of expressing a number or expression as a product of factors. Every positive integer has a unique prime factorization: n = p1^a1 p2^a2 ..., and the fundamental theorem of arithmetic states that this representation is unique up to the order of the factors. The number of positive factors of n can be computed from the exponents a1, a2, ... by the formula (a1+1)(a2+1)... For example, 360 = 2^3 × 3^2 × 5, and it has (3+1)(2+1)(1+1) = 4×3×2 = 24 positive factors.

In the context of polynomials, factoring means expressing a polynomial as a product of polynomials of lower degree. For instance, x^2 − 5x + 6 factors as (x−2)(x−3). Over a given field, polynomials can sometimes be factored further into irreducible components, with unique factorization up to order.

Beyond pure mathematics, "faktorer" also denote contributing elements in other disciplines, such as risk factors in health or factors in statistical analysis. In cryptography, factoring large integers is a hard problem that underpins the security of some cryptosystems, making efficient factorization algorithms an important area of study.