Prime factors of a number

Martin McBride
2022-01-27

Any integer n > 1 can be expressed as the product of one or more prime numbers. For example:

  • $30 = 2 \times 3 \times 5$
  • $63 = 3 \times 3 \times 7$
  • $48 = 2 \times 2 \times 2 \times 2 \times 3$

Those numbers are called the prime factors of n.

A prime number only has one prime factor - itself. For example, 3 is the only prime factor of 3.

Note that 1 is not a prime number, so we don't include it in the list of prime factors. The number 1 itself has no prime factors.

Multiplicity of prime factors

The number of times a prime number appears in the list of prime factors is called its multiplicity. For example, for the number 48:

  • Prime factor 2 has a multiplicity of 4.
  • Prime factor 3 has a multiplicity of 1.
  • We could also say that all the other prime numbers have a multiplicity of 0.

An alternative way to write prime factor is to use the multiplicity as a power, for example 48 is $2^4 \times 3^1$.

Unique factorization theorem

Any integer greater than 1 has a unique prime factorisation. In other words:

  • Every integer greater than 1 can be reoresented as a product of prime numbers.
  • There is only one way to represent that number as a product of prime numbers.

For example 63 is the product of the prime numbers 3, 3, and 7. There is no other way to produce 63 by multiplying primes. This is know as the unique factorization theorem (also called the fundamental theorem of arithmetic).