Finding the factors of a number is called **factorising**. In addition to the numbers 1 and itself, factors of a number can often be factorised themselves.

Any number can be factorised into a set of prime numbers: this is known as the **Product of the Prime Factors**, or the **Product of its Primes** (**Product** is another word for multiplication). The number 24

• can be written as 4 x 6,

• which can then be written as (2 x 2) x (2 x 3).

The Product of the Prime Factors for 24 is 2^{3} x 3.

A *Prime Factor Tree* can be used to obtain the Product of its Primes for a number. Start with the number. Create a pair of branches: at the end of each branch, enter any pair of factors for the number. Repeat the process for each branch unless it ends in a prime number. Multiply together the primes at the end of each branch.

90 = 2 x 5 x 3 x 3, re-arrange to:

90 = 2 x 3 x 3 x 5, re-write as powers

90 = 2 x 3^{2} x 5

Every number greater than 1 is either a prime number or can be represented as the product of its primes. A number will always factorize to the same set of primes, and that set of primes cannot be multiplied together for any different number (the **Unique Factorization Theorem**).

Write 48 as the product of its primes.

Answer: 48 = 2^{4} x 3.

Write 144 as the product of its primes.

Answer: 144 = 2^{4} x 3^{2}

See also Squares and Integer Powers and Roots

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