Factor

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 * 2 * 2 * 3 </math>

which is also written

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

The 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 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.