# Difference between revisions of "Factor"

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:<math>24 = 2^3 * 3</math>. | :<math>24 = 2^3 * 3</math>. | ||

− | + | The Prime Factorisation Theorem guarantees that every integer has one 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 |

## Revision as of 17:34, 14 May 2007

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.

The expression of an integer as a product of its prime factors is called a **prime factorization**. The prime factorization of 24 is

which is also written

- .

The Prime Factorisation Theorem guarantees that every integer has one unique prime factorization, e.g. 24 =2^{3}3^{1}, 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

and they are 1, 2, 3, 4, 6, 8, 12, and 24.