1. CHAPTER 5
Working With Number

2. Multiples
The MULTIPLES of a number are what you get when you multiply a number by positive whole number
eg 1 x 1 = 1 2 x 1 = 2 3 x 1 = 3
1 x 2 = 2 2 x 2 = 4 3 x 2 = 6
1 x 3 = 3 2 x 3 = 6 3 x 3 = 9
1 x 4 = 4 2 x 4 = 8 3 x 4 = 12
1 x 5 = 5 2 x 5 = 10 3 x 5 = 15
1 x 6 = 6 2 x 6 = 12 3 x 6 = 15
1 x 7 = 7 2 x 7 = 14 3 x 7 = 21
1 x 8 = 8 2 x 8 = 16 3 x 8 = 24 etc
The MULTIPLES of 1 are 1,2,3,4,5,6,7,8……..
The MULTIPLES of 2 are 2,4,6,8,10,12,14,16……
The MULTIPLES of 3 are 3,6,9,12,15,18,21…..
MULTIPLES is just another name for TIMES TABLES

3. Factors
The FACTORS of a number are whole numbers that divide exactly into a number
eg The factors of 2 are 1 and 2
The factors of 3 are 1 and 3
The factors of 4 are 1,2 and 4
The factors of 5 are 1 and 5
The factors of 6 are 1,2,3 and 6
The factors of 7 are 1 and 7
The factors of 8 are 1,2,4 and 8
The factors of 9 are 1,3 and 9
The factors of 10 are 1,2,5 and 10
etc

4. Prime Numbers
A PRIME NUMBER is a whole number greater than 1 which has only two factors 1 and itself eg 2,3,5,7,11,13,17,19,23,29,31,37……

5. Powers A POWER tells us how many of a number are multiplied together
eg 46 = 4 x 4 x 4 x 4 x 4 x 4
= 4096
The power a number is raised to is called an INDEX (plural INDICES) eg 95
122

6. Prime Factors Those FACTORS of a number which are PRIME NUMBERS
are called PRIME FRACTORS.
eg The factors of 24 are 1,2,3,4,6,8,12,24
So the PRIME FACTORS of 24 or 23
Products of Prime Factors
Any number can be written as a Product of its Prime Factors
eg Write 74 as a product of its prime factors

7. Highest Common Factor (HCF)
The HIGHEST COMMON FACTOR of two or more numbers
is the largest number that is a FACTOR of all the numbers.
To find the HCF of a set of numbers write them all as a
PRODUCT OF THEIR PRIME FACTORS and identify what is
common in all the lists.
eg Find the HCF of 18 and 45

8. Lowest Common Multiple (LCM)
The LOWEST COMMON MULTIPLE of two or more numbers
is the smallest number that is a MULTIPLE of all the numbers.
To find the LCM of a set of numbers write the first few
MULTIPLES of each of the numbers and look for the smallest
number that appears in all of the lists.
eg Find the LCM of 15 and 40

9. Square Numbers SQUARE NUMBERS are what you get when you multiply a
whole number by itself.
eg 12 = 1 x 1 = = 10 x 10 = 100
22 = 2 x 2 = = 11 x 11 = 121
32 = 3 x 3 = = 12 x 12 = 144
42 = 4 x 4 = = 13 x 13 = 169
52 = 5 x 5 = = 14 x 14 = 196
62 = 6 x 6 = = 15 x 15 = 225
72 = 7 x 7 = = 16 x 16 = 256
82 = 8 x 8 = = 17 x 17 = 289
92 = 9 x 9 = = 18 x 18 = 324 etc
1,4,9,16,25,36,49,64,81,100,121,144,169,…are SQUARE
NUMBERS

10. Cube Numbers CUBE NUMBERS are what you get when you multiply a
whole number by itself and then multiply by the number again
eg 13 = 1 x 1 x 1= = 6 x 6 x 6 = 216
23 = 2 x 2 x 2 = = 7 x 7 x 7 = 343
33 = 3 x 3 x 3 = = 8 x 8 x 8 =512
43 = 4 x 4 x 4 = = 9 x 9 x 9 = 729
53 = 5 x 5 x 5 = = 10 x 10 x 10 = 1000
etc
1,8,27,64,125,216,343,512,729,1000,….are CUBE
NUMBERS

11. Reciprocals The RECIPROCAL of a number is the value obtained when the
number is divided into 1.
So the RECIPROCAL of a number x is 1 x
A number multiplied by its RECIPROCAL always equals 1.
eg The RECIPROCAL of 2 is 1 2
so 2 x 1 = 1
The RECIPROCAL of a number is written with an INDEX of –1
so:
1 = 7-1 7

12. Square Roots When you find the SQUARE ROOT of a number you work out
what number was multiplied by itself to give the number under the
square root sign.
eg √81 = 9 because 9 x 9 = 81
Another way of writing a square root is to use an INDEX of ½
eg √64 = 64½ = 8 because 8 x 8 = 64

13. Cube Roots When you find the CUBE ROOT of a number you work out
what number was multiplied by itself twice to give the number under
the cube root sign.
eg 3√125 = 5 because 5 x 5 x 5 = 125
Another way of writing a cube root is to use an INDEX of ⅓
eg 3√27 = 27⅓ = 3 because 3 x 3 x 3 = 27

14. Multiplying Powers Of The Same Number
For example,
43 x 45 = (4x4x4) x (4x4x4x4x4)
= 4x4x4x4x4x4x4x4
43 x 45 = 48
so 43 x 45 = 43+5
= 48
When you MULTIPLY two powers of the same number
you ADD the indices. In general:
xa x xb = xa+b

15. Dividing Powers Of The Same Number
For example,
57 ÷ 54 = 5 x 5 x 5 x 5 x 5 x 5 x 5
5 x 5 x 5 x 5
= 5 x 5 x 5
57 ÷ 54 = 53
so 57 ÷ 54 = 57-4
= 53
When you DIVIDE two powers of the same number you
SUBTRACT the indices. In general
xa ÷ xb = xa-b
xa = xa-b xb

16. Surds
A SURD is the square root of a positive integer for
which the root is not a whole number.
Eg √2, √3, √5, √6 etc
We use SURDS when we want to keep an answer
EXACT.

17. Surds Rule 1 √a x b = √a x √b or √a x √b = √a x b eg eg
√32 = √16 x 2 = √16 x √2 = 4√2 √3 x √7 = √3 x 7 = √21
Rule 2
m√a + n√a = (m + n)√a or m√a - n√a = (m - n)√a
eg eg
2√2 + 3√ √3 - 5√3
5√ √3

18. Surds Rule 3 eg When simplifying surds look for FACTORS which
are SQUARE NUMBERS.