Factor (mathematics).

Factor (mathematics). Factor (mathematics), any one of two or more numbers multiplied together to form a product. The factors of any given whole number (numbers such as 1, 2, and 3 that can be evenly divided by 1) are themselves whole numbers by which the given number may be divided evenly--that is, without remainder. For example, since 6 is the product of 2 × 3, 2 and 3 are factors of 6. The decimal number 1.5 is not a factor of 6, because it is not a whole number, and 5 is not a factor of 6 because 6 cannot be divided evenly by 5. Since 1 × 6 = 6, 1 and 6 are also factors of 6. All whole numbers are factors of themselves, and 1 is a factor of all whole numbers. Some numbers such as 1, 2, 3, 5, and 7, have no factors other than themselves and one; these are prime numbers. Numbers that do have other factors are called composite numbers. The first five composite numbers are 4, 6, 8, 9, and 10. The process of breaking a number down into its factors other than 1 and itself is called factoring. For example, the composite number 72 can be factored 9 × 8. The numbers 9 and 8 are themselves composite numbers and can be further broken down into 3 × 3 and 2 × 2 × 2. Since 3 and 2 are prime and cannot be factored further, they are called the prime factors of 72. All composite numbers can be expressed as products of prime numbers. Factoring very large numbers is difficult, but the problem is of current interest because of its use in designing secure encryption (see Cryptography). Two or more numbers may share several factors. For example, both 18 and 24 can be divided by 1, 2, 3, or 6. These numbers are called the common factors of 18 and 24. The number 6 is the largest of these common factors and is therefore called the highest common factor (HCF) of 18 and 24. The highest common factor is used in a number of important mathematical operations. For example, all fractions can be reduced to their lowest terms by dividing both numerator and denominator by their highest common factor: Also, when two or more fractions must be multiplied, extensive calculations can often be avoided by finding the HCF of each fraction. For example: In algebra, expressions can be factored like numbers. The factors of an algebraic expression are themselves expressions that, when multiplied together, produce the original expression. For example, the factors of the expression x2 - y2 are (x - y) and (x + y) because (x - y)(x + y) = x2 - y2. These are the only factors that can be found for x2 - y2. Factoring is often an indispensable step in solving algebraic problems. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.