Prime Numbers

PRIME NUMBERS

A prime number is a number whose only divisors are 1 and itself. In ascending order, the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc etc.

Note that by convention 1 is not a prime number.

Prime numbers are the building blocks of arithmetic: every number can be expressed as a product of prime numbers ie prime numbers multiplied together.

Example

Find the prime factors of 468.

Start with the first prime number, 2. Is 468 divisible by 2? Yes:

468 = 234 x 2

Now move to 234. Is 234 divisible by 2? Yes:

234 = 117 x 2

Now move to 117. Is 117 divisible by 2? No. Go to the next prime number, 3. Is 117 divisible by 3? Yes:

117 x 39 x 3

Now move to 39. Is 39 divisible by 2. No. Go to the next prime number, 3. Is 39 divisible by 3. Yes:

39 = 13 x 3

Now move to 13 and repeat. Is 13 divisible by 2? No. By 3. No. By the next prime, 5? No. By the next prime, 7? No. 13 is a prime number so we stop.

We can now collect the prime numbers at each stage - 2, 2, 3, 3 and 13 and write

468 = 2 x 2 x 3 x 3 x 13.

This is known as the Factor Tree. Factor trees aren`t difficult but they are difficult to show here - find any text book and see how the above solution can be set out as a `tree`. Or just ask me!

You can use factor trees to work out the HCF and LCMs of two numbers.

Example

Find the HCF and LCMs of 24 and 64.

Using the factor tree approach above

24 = 3 x 2 x 2 x 2

64 = 2 x 2 x 2 x 2 x 2 x 2

If we circle the numbers common to both lists then multiply them together we get

HCF = 2 x 2 x 2 = 8

For the LCM, write out all the numbers from both trees but where there are duplicates only include that number once. And then multiply them together. So

LCM = 3 x 2 x 2 x 2 x 2 x 2 x 2 = 192.

Note that if you multiply the HCF and the LCM of two numbers by each other you get the product of the two numbers.