Hence, the prime factors of 100 are 1, 2, 2, 5, 5, and 1 × 2 × 2 × 5 × 5 = 100. From the above illustration, we derive the following rule for finding all the prime factors of any composite number.

Theorem 5.5 Dickman's Theorem. The probability of a number N chosen at random having a prime factor between N^α and N) is approximately =δ, independent of the magnitude of α, if δ is small.

If any subset of the set of prime factors of a be taken, then the product of the elements of this subset is a factor of a.
For example, 12 is a factor of 60 and the set of prime factors of 12 is {2, 2, 3}, which is a subset of {2, 2, 3, 5}, which is the set of prime factors of 60.

Any composite number can be prime factored; that is, it can be written as a product of prime numbers (excluding 1) in one and only one way.[…]When prime factoring numbers, it is standard to rearrange the factors so that the numbers are in increasing order.

For example, the number 30, which is not prime, can be prime factored as 2 x 3 x 5.

Another variation of the technique for changing radicals to simplest form is to prime factor the radicand and then to look for perfect squares in exponential form.

Prime factor

Meaning

Noun

Verb