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 numbers.
The expression of an integer as a product of its prime factors is called a prime factorization. The prime factorization of 24 is
- 24 = 2 * 2 * 2 * 3
which is also written
- 24 = 23 * 3.
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
- (3 + 1) * (1 + 1) = 8
and they are 1, 2, 3, 4, 6, 8, 12, and 24. To show this in general, suppose that n = 2a 3b 5c. Then the factors of n are exactly the numbers 2j 3k 5l, 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).
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.