Changes

==Prime factorization==

==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>

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

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

:<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 =2³3¹, 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 =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

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>

:<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. To show this in general, suppose that n = 2^a 3^b 5^c.  Then the factors of n are exactly the numbers 2^j 3^k 5^l, 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==

==Other uses==

The word factor also appears in other contexts, such as algebra. A [[polynomial]], such as x³+3x²+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, x³+3x²+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 x³+3x²+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, x³+3x²+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:Arithmetic]]