# Brush up your maths: Numbers

## Factors

The factors of a number are all the whole numbers that divide into it. For example, 1, 2, 4, 5, 10, and 20 all divide into 20 exactly. We call these the factors of 20.

### Exercise: factors

What are the factors of 12?

## Prime numbers

A prime number is one which has only two factors: 1 and itself. For example, 7 is a prime number, because only 1 and 7 divide into it.

Sometimes it is useful to write a number as the multiple or product of all its prime factors. For example:

30 = 2 × 3 × 5

We usually omit the factor 1, as it doesn't really contribute anything. If a prime factor appears twice in a number, we include it twice:

18 = 2 × 3 × 3

27 = 3 × 3 × 3

### Exercise: Prime numbers

Which of these are prime numbers: 2, 3, 4, 5, 6, 31, 32, 33?

What are 42 and 24 as the products of their prime factors?

## Common factors and common multiples

The common factors of two numbers are all the whole numbers that divide into both of them. For example, 1 and 5 both divide into 15 and 20 exactly, nothing else does. The highest common factor is the largest number which will divide into both of them. For 15 and 20, the highest common factor is 5.

The lowest common multiple of two numbers is the smallest number of which they are both factors. For 15 and 20, this is 60. We find this by finding the highest common factor, 5. We divide this into each of the numbers: 15/5 = 3, 20/5 = 4. Of course, 3 and 4 have no common factors, or 5 would not be the highest common factor of 15 and 20. We then multiply the highest common factor by the two other factors, which are not common: 5 × 3 × 4 = 60. This is the lowest common multiple.

The process is sometimes easier if we first express each number as the product of its prime factors. For example, what is the lowest common multiple of 8 and 28?

8 = 2 × 2 × 2

28 = 2 × 2 × 7

The highest common factor is 2 × 2 = 4. It is easy to see that when we remove 2 × 2 from the lists of factors, 8 ÷ 4 = 2, 28 ÷4 = 7. Hence the lowest common multiple is 2 × 2 × 2 × 7 = 56.

### Exercise: common factors

What are the common factors of 12 and 18 and what is their highest common factor? What is their lowest common multiple?

## Brackets

We use brackets to change the order in which we do parts of a calculation. For example, we calculate

4 × 7 − 2

by first multiplying 4 by 7 to give 28 and then subtracting 2 to give 26.

4 × 7 − 2 = 28 − 2 = 26

If we want to subtract 2 from 7 then multiply by 4 we put brackets round the 7 − 2, like this:

4 × (7 − 2)

Now we subtract 2 from 7 to give 5 and then multiply 5 by 4 to give 20.

4 × (7 − 2) = 4 × 5 = 20.

### Exercise: brackets

Calculate the following:

(9 + 1) × (5 − 2)

(8 + 4) ÷ (5 − 1)

## The order of operations

A handy mnemonic for the order of operations is "BODMAS":
Brackets
Of (which we think of as "to the power of", more later)
Divide
Multiply
Subtract

For example, how do we calculate

4 × (7 − 3) / (5 + 3) + 8 ?

(N.B. "/" means "divide", 6/2 = 6 ÷ 2 = 3.)

We first calculate the brackets (7 − 3) = 4 and (5 + 3) = 8 to give

4 × 4 / 8 + 8

We now divide 4 by 8 to give ½ and so we get:

4 × ½ + 8

Now we multipy 4 by ½ to give 2 and we get:

2 + 8 = 10.

### Exercise: BODMAS

Calculate the following:

(9 + 1) × (5 − 2) + 8/2 − 3

## Negative numbers

In the previous exercises we have done some subtractions, one number minus another, e.g. 7 − 2 = 5. A negative number is one that carries its minus sign with it.

For example, −2 is a negative number. Adding −2 is the same as subtracting 2:

−2 + 7 = 7 − 2 = 5.

Note that −2 is a smaller number than −1. It is more negative. When we add −2 to smoething, we end up with less than we do when we add −1.

We can see this if we arrange the numbers in a number line:

smaller ←   . . .   −10   −9   −8   −7   −6   −5   −4   −3   −2   −1   0   1   2   3   4   5   6   7   8   9   10   . . .   → bigger

If we mutliply a negative number by a positive number we get a negative number: −2 × 7 = −14

This is because −2 times 7 is like taking away 2 seven times, i.e. taking away 14.

If we mutlipy a negative number by another negative number we get a positive number: −2 × −7 = 14

Taking 2 away minus 7 times is like giving 2 plus 7 times.

We sometimes put brackets round negative numbers to make clearer what we are doing:

(−2) × (−7) = 14.

### Exercise: negative numbers

Calculate the following:

(−10) × (5 − 2)

(−10) × (2 − 5)

A lottery scratch game would be won if the card revealed a temperature lower than −8 degrees centrigrade. A gambler bought a card which revealed −6 degrees. She thought she had won, the shopkeeper thought she had won, but Camelot thought she had lost. Who was right?

## Absolute values

If we have a negative number, such as −2, sometimes we want to use the number without its minus sign, i.e. as 2. We write it with two vertical lines round it:

|−2| = 2.

The absolute value of a positive number is just the number itself:

|2| = 2.

The absolute values lines also act like a bracket, so

|2 − 5| × 2 = 6.

We get |2 − 5| = |−3| = 3.

### Exercise: absolute values

Calculate the following:

|−10| × |2 − 5|

## Squares and square roots

Square numbers are formed by multiplying a whole number by itself, e.g. 5 × 5 =25 and 25 is a square number. We also call this a perfect square.

We can write it like this: 52 = 25. The expression 52 means two fives multiplied together. We can call it 5 squared, 5 raised to the power of 2, or 5 to the 2.

5 is the square root of 25, because 5 squared = 25. We write this as 5 = √25.

The square of a negative number is positive, because it is two negative numbers multiplied together:

(−5)2 = (−5) × (−5) = 25.

This means that 25 has two square roots: 5 and −5. We sometimes write this as √25 = ±5. The symbol "±" means "plus or minus".

Negative numbers have no square roots, outside the world of the imagination.

### Exercise: squares and square roots

Calculate the following:

32 + (−2)2

√144

√(32 + 42)