Factor - Revision history

Abcqwe: /* Other uses */ Recat

2017-03-26T13:38:08Z

Other uses: Recat

DRBSouthington: /* Prime factorization */

2009-11-29T17:15:15Z

Prime factorization

MarkGall: added an explanation

2009-09-29T00:04:10Z

added an explanation

Foxtrot: cat

2008-11-16T11:40:07Z

cat

Foxtrot: consistent spelling, fixed link

2008-07-14T16:59:58Z

consistent spelling, fixed link

Foxtrot: adding sections to allow a sublink

2008-07-14T03:10:37Z

adding sections to allow a sublink

Icspeks at 02:29, July 14, 2008

2008-07-14T02:29:41Z

TruthinessBee at 19:25, November 15, 2007

2007-11-15T19:25:01Z

Fox at 09:49, May 25, 2007

2007-05-25T09:49:36Z

Fox: Undo revision 176400 by Special:Contributions/A Noble Ox (User talk:A Noble Ox) removing vandalism

2007-05-25T09:49:27Z

Undo revision 176400 by Special:Contributions/A Noble Ox (User talk:A Noble Ox) removing vandalism

New page

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 factorisation of 24 is

:<math>24 = 2 * 22 * 2 * 3 </math>

which is also written

:<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

:<math>(3 + 1) * (1 + 1) = 8</math>

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

[[Category:Mathematics]]

← Older revision		Revision as of 13:38, March 26, 2017
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	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]].		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:~~Number Theory~~]]	+	[[Category:Arithmetic]]

@@ Line 19: / Line 19: @@
 :<math>(3 + 1) * (1 + 1) = 8</math>
-and they are 1, 2, 3, 4, 6, 8, 12, and 24.  It's easy to see why this formula works.  Suppose, for example, that n = 2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup>.  Then the factors of n are exactly the numbers 2<sup>j</sup> 3<sup>k</sup> 5<sup>l</sup>, where j is between 0 and a, k is between 0 and b, and l is between 0 and c.  So the number of factors is just the number of choices for j, times the number of choices for k, times the number of choices for l, which is (a+1)(b+1)(c+1).
+and they are 1, 2, 3, 4, 6, 8, 12, and 24. To show this in general, suppose that n = 2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup>.  Then the factors of n are exactly the numbers 2<sup>j</sup> 3<sup>k</sup> 5<sup>l</sup>, where j is between 0 and a, k is between 0 and b, and l is between 0 and c.  So the number of factors is just the number of choices for j, times the number of choices for k, times the number of choices for l, which is (a+1)(b+1)(c+1).
 ==Other uses==

@@ Line 19: / Line 19: @@
 :<math>(3 + 1) * (1 + 1) = 8</math>
-and they are 1, 2, 3, 4, 6, 8, 12, and 24.
+and they are 1, 2, 3, 4, 6, 8, 12, and 24.  It's easy to see why this formula works.  Suppose, for example, that n = 2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup>.  Then the factors of n are exactly the numbers 2<sup>j</sup> 3<sup>k</sup> 5<sup>l</sup>, where j is between 0 and a, k is between 0 and b, and l is between 0 and c.  So the number of factors is just the number of choices for j, times the number of choices for k, times the number of choices for l, which is (a+1)(b+1)(c+1).
 ==Other uses==

← Older revision		Revision as of 11:40, November 16, 2008
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	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]].		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:Number Theory]]

@@ Line 5: / Line 5: @@
 ==Prime factorization==
-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 factorization of 24 is
 :<math>24 = 2 * 2 * 2 * 3 </math>
@@ Line 13: / Line 13: @@
 :<math>24 = 2^3 * 3</math>.
-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 [[Prime Factorization 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

@@ Line 2: / Line 2: @@
 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
@@ Line 18: / Line 20: @@
 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]]

@@ Line 11: / Line 11: @@
 :<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 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