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CHAPTER 5 Working With Number

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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

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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

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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……

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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.

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**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

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**Surds Rule 3 eg When simplifying surds look for FACTORS which**

are SQUARE NUMBERS.

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