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Line 1: − A '''factor''' is an [[integer]] that evenly divides another integer. For example, 3 is a factor of 24 because 24 divided by 3 does not leave a remainder. 5 is not a factor of 24.+ − Factors are sometimes called '''divisors''' to distinguish them from '''prime factors'''. A prime factor is a divisor that is a [[prime number]]. 2 and 3 are prime factors of 24. 6 is not a prime factor because it is a [[composite number]].+ Line 11:
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An [[integer]] '''factor''' or '''divisor''' is an integer that evenly divides another integer, so the ratio is also an integer. For example, 3 is a factor of 24 because 24 divided by 3 is 8, which is an integer. Five is not a factor of 24, because 24 divided by 5 is the [[decimal]] 4.8, or the [[mixed fraction]] 4 4/5, which is not an integer.
A prime factor is a factor that is a [[prime number]]. The prime factors of 24 are 2 and 3. The other positive factors of 24 are 1, 4, 6, 8, 12, and 24, but these are not prime, but are [[composite number]]s.
The expression of an integer as a product of its prime factors is called a '''prime factorization'''. The prime factorisation of 24 is
The expression of an integer as a product of its prime factors is called a '''prime factorization'''. The prime factorisation of 24 is
:<math>24 = 2^3 * 3</math>.
:<math>24 = 2^3 * 3</math>.
The Prime Factorisation Theorem guarantees that every integer has a unique prime factorization, e.g. 24 =2<sup>3</sup>3<sup>1</sup>, though it may have multiple non-prime factorizations (e.g. 24 = 2 * 12, 6 * 4, 3 * 8).
The [[Fundamental Theorem of Arithmetic|Prime Factorisation Theorem]] guarantees that every integer has a unique prime factorization, e.g. 24 =2<sup>3</sup>3<sup>1</sup>, though it may have multiple non-prime factorizations (e.g. 24 = 2 * 12, 6 * 4, 3 * 8).
The number of divisors of an integer may be determined from its prime factorization when expressed in exponent form, by incrementing each exponent by 1 and multiplying the results. In the example above, the exponents of prime factors 2 and 3 are 3 and 1, respectively. The number of divisors of 24 is therefore
The number of divisors of an integer may be determined from its prime factorization when expressed in exponent form, by incrementing each exponent by 1 and multiplying the results. In the example above, the exponents of prime factors 2 and 3 are 3 and 1, respectively. The number of divisors of 24 is therefore
and they are 1, 2, 3, 4, 6, 8, 12, and 24.
and they are 1, 2, 3, 4, 6, 8, 12, and 24.
The word factor also appears in other contexts, such as algebra. A [[polynomial]], such as x<sup>3</sup>+3x<sup>2</sup>+2x, may be decomposed into a product of linear terms (terms of the form (ax+b) where a and b are real numbers, and a is nonzero); for example, x<sup>3</sup>+3x<sup>2</sup>+2x=x(x+1)(x+2). These linear terms are called the factors of the polynomial, and the process of determining the factors is called [[factorization]].
[[Category:Mathematics]]
[[Category:Mathematics]]