Difference between revisions of "Factor"

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:<math>24 = 2^3 * 3</math>.
 
:<math>24 = 2^3 * 3</math>.
  
Every integer has one unique prime factorization, though it may have multiple non-prime factorizations (e.g. 24 = 2 * 12, 24 = 3 * 8).
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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 =2331, 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.