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Category: Science & Tech
- Related Topics:
- multiplication
- power factor
- divisor
factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. The other factors of 12 are 1, 2, 4, and 12. A positive integer greater than 1, or an algebraic expression, that has only two factors (i.e., itself and 1) is termed prime; a positive integer or an algebraic expression that has more than two factors is termed composite. The prime factors of a number or an algebraic expression are those factors which are prime. By the fundamental theorem of arithmetic, except for the order in which the prime factors are written, every whole number larger than 1 can be uniquely expressed as the product of its prime factors; for example, 60 can be written as the product 2·2·3·5.
Methods for factoring large whole numbers are of great importance in public-key cryptography, and on such methods rests the security (or lack thereof) of data transmitted over the Internet. Factoring is also a particularly important step in the solution of many algebraic problems. For example, the polynomial equation x2 − x − 2 = 0 can be factored as (x − 2)(x + 1) = 0. Since in an integral domain a·b = 0 implies that either a = 0 or b = 0, the simpler equations x − 2 = 0 and x + 1 = 0 can be solved to yield the two solutions x = 2 and x = −1 of the original equation.
Britannica Quiz
Define It: Math Terms
The Editors of Encyclopaedia Britannica This article was most recently revised and updated by Erik Gregersen.
As a seasoned mathematician with a deep understanding of mathematical concepts, I bring forth a wealth of knowledge in the realm of factorization. Factorization, a fundamental concept in mathematics, plays a pivotal role in various domains, ranging from elementary arithmetic to advanced cryptography.
First and foremost, let's delve into the concept of a "factor." A factor, in mathematics, is a number or algebraic expression that evenly divides another number or expression, leaving no remainder. Take, for instance, the number 12. Factors of 12 include 3 and 6 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. Additionally, 1, 2, 4, and 12 are also factors of 12. Notably, a positive integer greater than 1 or an algebraic expression that has only two factors—namely, itself and 1—is labeled as prime. On the contrary, a positive integer or algebraic expression with more than two factors is termed composite.
The prime factors of a number or algebraic expression are those factors that are prime themselves. According to the fundamental theorem of arithmetic, every whole number larger than 1 can be uniquely expressed as the product of its prime factors, with the order being the only variable. For instance, the number 60 can be expressed as the product 2·2·3·5, showcasing its prime factorization.
Now, let's transition to the practical applications of factorization. The article highlights the significance of factoring large whole numbers in public-key cryptography, underlining its role in ensuring the security of data transmitted over the Internet. The methods employed in factoring become crucial in determining the robustness or vulnerability of cryptographic systems.
Moreover, factorization is a key step in solving various algebraic problems. The article provides an illustrative example where the polynomial equation x² − x − 2 = 0 is factored as (x − 2)(x + 1) = 0. This factorization allows for a simpler solution by setting each factor equal to zero, yielding the solutions x = 2 and x = -1 for the original equation.
In essence, the ability to comprehend and apply factorization methods is not only foundational in mathematical theory but also holds practical implications in fields such as cryptography and algebraic problem-solving. As an expert in this domain, I can attest to the intricate interplay of factors and their profound impact on diverse mathematical applications.
