# Prime Factorisation

## (how to write a whole number as a product of its prime factors)

In this section we learn about prime factorisation. In other words we learn how to write a whole number as a product of its prime factors.

Given a whole number $$n$$, a prime factor of $$n$$ is a factor of $$n$$ which is also a prime number.

The first few prime numbers are: $2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19, \dots$ Every whole number, greater than $$1$$, is either one of the prime numbers or can be written as a product of prime numbers. For instance, $$18$$ can be written: $18 = 2\times 3^2$ The fact that this type of factorisation can be done for all whole numbers greater than $$1$$ stems from the fundamental theorem of arithmetic.

### Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic states:

Every integer greater than $$1$$ is either a prime number or can be written as a product of its prime factors.

This means that every whole number, that is greater than $$1$$ can be written as a product of its prime factors (no exceptions). The method for doing this is explained in the following tutorial.

## Exercise 1

Write each of the following whole numbers as its product of prime factors:

1. $$36$$
2. $$90$$
3. $$196$$
4. $$900$$
5. $$384$$
6. $$3600$$
7. $$210$$
8. $$540$$
9. $$144$$
10. $$180$$

Note: this exercise can be downloaded as a worksheet to practice with:

## Solution Without Working

We find the following results:

1. $$36 = 2^2\times 3^2$$
2. $$90 = 2\times 3^2\times 5$$
3. $$196 = 2^2\times 7^2$$
4. $$900 = 2^2\times 3^2 \times 5^2$$
5. $$384 = 2^7\times 3$$
6. $$3600 = 2^4 \times 3^2 \times 5^2$$
7. $$210 = 2\times 3\times 5\times 7$$
8. $$540 = 2^2 \times 3^3 \times 5$$
9. $$144 = 2^4 \times 3^2$$
10. $$180 = 2^2 \times 3^2 \times 5$$