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Mathematics

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Published in: Mathematics
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Area,volume and surface formulas,conic section,algebra,numerical and complex,

Arman R / Rajkot

3 years of teaching experience

Qualification: M.Sc (The M S University of Baroda - 2017), B.Sc (The M S University of Baroda - 2015)

Teaches: Algebra, Mathematics

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  1. 2 Factors Factors and Prime Numbers 2.1 A factor divides exactly into a number, leaving no remainder. For example, 13 is a factor of 26 because 26 + 13 = 2 leaving no remainder. A prime number has only two factors, 1 and itself; this is how a prime number is defined. 5 is a prime number because it has only two factors, 1 and 5. 8 has factors 1, 2, 4 and 8, so it is not prime. 1 is not a prime number because it has only one factor, namely 1 itself. Example 1 (a) List the factors of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. (b) Which of these numbers are prime numbers? Solution 1, 1, 1, 1, 1, 1, 2, 5, 10 (a) (b) This table lists the factors of these numbers: Number 1 2 3 4 5 6 7 8 9 10 Factors 1 3 2, 5 7 2, 3, 4 9 The numbers 2, 3, 5 and 7 have exactly two factors, and so only they are prime numbers. 27
  2. 2.1 Example 2 List the prime factors of 24. Solution MEP Y8 Practice Book A First list all the factors of 24, and they are: 1, 2, 3, 4, 6, 8, 12, 24 Now select from this list the numbers that are prime; these are 2 and 3, and so the prime factors of 24 are 2 and 3. Example 3 Which of the following numbers are prime numbers: 18, 45, 79 and 90 ? Solution The factors of 18 are 1, 2, 3, 6, 9 and 18; 18 is not a prime number. The factors of 45 are 1, 3, 5, 9, 15 and 45; 45 is not a prime number. The factors of 79 are 1 and 79; 79 is a prime number The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90; 90 is not a prime number. 79 is the only prime number in the list. Divisibility Test If a number is divisible by 2, If a number is divisible by 3, If a number is divisible by 4, If a number is divisible by 5, If a number is divisible by 9, it will end with 0, 2, 4, 6 or 8. the sum of its digits will be a multiple of 3. the last two digits will be a multiple of 4. it will end in 0 or 5. the sum of its digits will be a multiple of 9. If a number is divisible by 10, it will end in 0. Can you find tests for divisibility by other numbers? 28
  3. MEP Y8 Practice Book A Exercises 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. (a) List all the factors of each of the following numbers: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (b) Which of these numbers are prime? Explain why 99 is not a prime number. Which of the following are prime numbers: 33, 35, 37, 39? Find the prime factors of 72. (a) Find the prime factors of 40. (b) Find the prime factors of 70. (c) Which prime factors do 40 and 70 have in common? Find the prime factors that 48 and 54 have in common. A number has prime factors 2, 5 and 7. Which is the smallest number that has these prime factors? The first 5 prime numbers are 2, 3, 5, 7 and 11. Which is the smallest number that has these prime factors? Write down the first two prime numbers which are greater than 100. Which is the first prime number that is greater than 200? 29
  4. 2.2 MEP Y8 Practice Book A Prime Factors A factor tree may be used to help find the prime factors of a number. Example 1 Draw a factor tree for the number 36. Solution Start with 36 and then: split 36 into numbers 9 and 4 that multiply to give 36 as shown in the factor tree opposite; repeat for the 9 and the 4, as shown on the factor tree. The factor tree is now complete because the numbers at the ends of the branches are prime numbers; the prime numbers have been ringed. Another possible factor tree for 36 is shown here: On the factor tree we only put a ring around the prime numbers. Note that, at the end of the branches, both the numbers 2 and 3 appear twice. The prime factors of 36 are 3, 2, 2 and 3. In ascending order, the prime factors of 36 are 2, 2, 3, 3. 36 9 36 12 6 4 From the factor trees above it is possible to write: 36 = When a number is written in this way, it is said to be written as the product of its prime factors. 30
  5. MEP Y8 Practice Book A Example 2 Express each of the following numbers as the product of its prime factors: 102 (a) Solution (b) 60 (a) (b) Start by creating a factor tree: 102 Start by creating a factor tree: 60 Put the prime numbers in ascending order: 60 102 3 5 51 17 60 15 4 Example 3 A number is expressed as the product of its prime factors as What is the number? Solution 360 31
  6. 2.2 Exercises MEP Y8 Practice Book A 1. 2. 3. 4. 5. 6. 7. 8. Draw factor trees for the following numbers: (a) 20 (b) 100 (c) 88 Draw two different factor trees for 40. (a) Draw two different factor trees for 66. (b) Can you draw any other different factor trees for 66? Copy the factor tree opposite and fill in the missing numbers: 12 Fill in the missing numbers on a copy of the factor tree opposite: 4 c Use a factor tree to find the prime factors of: (a) 30 (b) 80 2 (c) 200 Write each of the following numbers as the product of their prime factors: (a) 62 (d) 320 (g) 54 (b) 64 (e) 90 (h) 38 (c) 82 (D 120 (i) 1000 A number is expressed as the product of its prime factors as: What is the number? 32
  7. 9. 10. 11. MEP Y8 Practice Book A The prime factors of a number are 2, 7 and 11. Which are the three smallest numbers with these prime factors? Which is the smallest number that has: (a) (b) (a) (b) 4 different prime factors, 5 prime factors? Write down two numbers, neither of which must end in 0, and which multiply together to give 1000. Repeat question 11 (a), this time writing down two numbers which multiply to give 1 000 000. 2.3 Index Notation You will have seen the occasional use of index notation in the last section; for example, in the statement 23 x 32 x 5 which contains 2 indices. = 360 We read 23 as "two to the power of three" or "two cubed": 2 is the base number, 3 is the index. In general, an is the result of multiplying the base number, a, by itself n times, n being the index. a = a x ax ax xaxaxaxa A calculator can be used to work out powers. The index button is usually marked or yx. Sometimes you will need to press the SHIFT or 2nd FUNCTION key before using the index button. You should find out which buttons you need to use on your calculator. For example, to calculate 54 you may need to press either or to get the correct answer of 625. 33
  8. 2.3 Example 1 Calculate: (a) 24 MEP Y8 Practice Book A (b) 73 Check your answers using a calculator. Solution (c) Y (a) (b) (c) 24 73 105 16 Using a calculator, either or 343 Using a calculator, either or 100 000 Using a calculator, either or 105 16 16 343 343 100 000 100 000 Example 2 Write these statements, filling in the missing numbers: (a) 32 = 2 (b) 1000000 = 10 Solution (a) (b) 32 25 1 000 000 106 34
  9. MEP Y8 Practice Book A Exercises 1. 2. 3. 4. 5. Copy the following statements and fill in the missing numbers: (b) (d) Calculate: (a) (d) (g) 23 53 92 (b) (h) 33 27 103 (c) (D 104 34 7 10 Copy the following statements and fill in the missing numbers: (a) 100 = 10 (d) 16 — C] Calculate: (a) 52 x 22 (d) 62 x 2 (b) (b) 81 = 16 = 32 x 24 92 x 3 (c) (c) (f) 4 - 2401 72 x 23 53 x 23 Copy each of the following statements and fill in the missing numbers: (a) (b) (c) (d) 23 x 25 = x 64 >< 62 73 x 77 35
  10. 2.3 = 83 4 = 284 59 x 2.4 MEP Y8 Practice Book A Copy the following statements and fill in the missing numbers: 6. 7. 8. 9. 10. (a) 93 (b) 94 If Y, which number does n represent? Answer this question using a similar method to the one used in question 6. If 4n = 212, which number does n represent? If 1254 = Y, which number does n represent? Copy the following statements and fill in the missing numbers: (a) (c) 54 4 x 23 74 x (b) (d) (f) 5 75 x 35 9 = 109 Highest Common Factor and Lowest Common Multiple The highest common factor (HCF) of two numbers is the largest number that is a factor of both. The factors of 12 are 1, 2, The factors of 15 are 1, 3, so the HCF of 12 and 15 is 3. 3, 5, 12. 15. The HCF is easy to find for some numbers, but for others it is more difficult. In harder cases, the best way to find the HCF is to use prime factors. 36
  11. Example 1 Find the HCF of: (a) 20 and 30 Solution MEP Y8 Practice Book A = 12 (b) 2, 2, 14 and 12 (a) (b) The factors of 20 are 1, The factors of 30 are 1, 4, 3, 7 3, 5, 5, 10 6, and 20. 10, 15 and 30. The HCF of 20 and 30 is 10. The factors of 14 are 1, 2, The factors of 12 are 1, 2, The HCF of 14 and 12 is 2. Example 2 Find the HCF of 60 and 72. Solution Using factor trees: 60 30 15 5 60 and 14. 4, 6 and 12. 36 72 9 18 3 72 2 23 x 32 The HCF is calculated using the prime factors that are common to both numbers. In this case, 2 appears twice in both, and 3 appears once in both. so, the HCF of 60 and 72 37
  12. 2.4 60 — 60 — 72 - 72 - HCF = 2 x 2 HCF = 12 MEP Y8 Practice Book A To be in the HCF, the prime factor must be in both lists: Alternatively, using indices: 5 2 x x >< 3 22 x 31 x 51 x x Lowest power of 3 HCF 22 x 31 x 50 Lowest power of 2 Lowest power of 5 HCF 12 The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both. For example, 18 is the smallest number that is a multiple of both 6 and 9, so the LCM of 6 9 is 18. Example 3 What is the LCM of: (a) 5 and 7 Solution (b) 6 and 10 (a) (b) The multiples of 5 are: 5, 10, 15, 20, 25, 30, The multiples of 7 are: 7, 14, 21, 28, 35, 42, 35, 49, 40, 38 45, The LCM of 5 and 7 - 35. The multiples of 6 are: 6, 12, 18, 24, 30, 36, The multiples of 10 are: 10, 20, 30, 40, 50, The LCM of 6 and 10 60, - 30.
  13. MEP Y8 Practice Book A The LCM for larger numbers can be found by using prime factorisation. Example 4 Find the LCM of 60 and 72. 60 — 72 - LCM = 2 x LCM = 360 Solution From Example 2, 60 and 72 23 x 32 The LCM includes all the factors from either number. To be in the LCM, the prime factor can be in either list or in both lists: Alternatively, using indices: 60 — 72 - Example 5 x 2 x x 2 x x 3 x 3 x >< 5 LCM - Highest power of 2 31 x 51 23 x 50 Highest power of 3 23 x 32 x Highest power of 5 LCM = 360 Find the HCF and LCM of 50 and 70. Solution Using factor trees to find the prime factorisations: 50 50 25 21 x 52 7 70 39 70 10 2 21 x 51 x 71
  14. 2.4 HCF LCM Exercises 21 x 51 x 70 10 21 x 71 350 MEP Y8 Practice Book A HCF 50 70 LCM >< 5 5 1. 2. 3. 4. 5. 6. 7. (a) List the factors of 21. (b) List the factors of 35. (c) What is the HCF of 21 and 35 ? Find the HCF of: (a) (c) (a) (b) (c) 6 and 9 30 and 24 (b) (d) 14 and 18 15 and 10 Use a factor tree to find the prime factorisation of 42. Use a factor tree to find the prime factorisation of 90. Find the HCF of 42 and 90. What is the HCF of: (a) (d) (a) (b) (c) 90 and 120 77 and 50 (b) 96 and 72 300 and 550 List the first 10 multiples of 8. List the first 10 multiples of 6. What is the LCM of 6 and 8 ? What is the LCM of: (b) 9 and 6 15 and 20 (c) (c) (D 56 and 60 320 and 128 ? 8 and 10 6 and 11 ? (a) (d) (a) (b) (c) 5 and 3 12 and 9 Use a factor tree to find the prime factorisation of 66. Use a factor tree to find the prime factorisation of 40. Find the LCM of 40 and 66. 40
  15. MEP Y8 Practice Book A 8. 9. 10. Find the LCM of: (a) 28 and 30 (d) 60 and 50 (b) 16 and 24 12 and 18 (c) 20 and 25 21 and 35 Two lighthouses can be seen from the top of a hill. The first flashes once every 8 seconds, and the other flashes once every 15 seconds. If they flash simultaneously, how long is it until they flash again at the same time? At a go-kart race track, Vic completes a lap in 40 seconds; Paul completes a lap in 30 seconds, and Mark completes a lap in 50 seconds. If all three start a lap at the same time, how long is it before (a) Paul overtakes Vic, (b) Vic overtakes Mark? Squares and Square Roots 2.5 To square a number you multiply the number by itself. If you square 8, you multiply 8 by 8: — 64 so the square of 8 64. 2 The calculator button for squaring numbers usually looks like x or . For the second type of calculator you have to press the SHIFT or 2nd FUNCTION key first. Sometimes we need to answer questions such as, "What number was squared to get 64?" When answering this we need to use square roots. The square root of a number is a number which, when squared (multiplied by itself), gives you the first number. The sign means square root. 64 = 8 We say that: the square root of 64 is 8, i.e. since the square of 8 is 64, i.e. 82 41 = 64
  16. MEP Y8 Practice Book A 2.5 The calculator button for finding a square root usually looks like With some calculators you press the square root button before entering the number; with others you enter the number and then press the square root button. You need to find out how your calculator works out square roots. Example 1 (a) Square each of these numbers: 1, 5, 7, 14 52 = 25 72 = 49 12=1 (b) Find: 25, Solution 49 , 14 X 14 5 7 14 1 196, = 25 = 49 1 (a) (b) 12 52 72 142 25 49 196 — 1 = 196 because because because because 142= 196 Example 2 Use your calculator to find: (a) 542 (b) Solution (a) Either @ @ @ @ 2 2916 (b) Either @ 31 or C @ @ @ 31 42
  17. MEP Y8 Practice Book A Example 3 2 The area of this square is 225 cm What is the length of each side? Solution Area 2 225 cm 2 (length of side) 2 — (length of side) Length of side Example 4 225 15 cm Use your calculator to find Solution 5 = 2.236067977 5 correct to 2 decimal places. = 2.24 correct to 2 decimal places. Exercises 20 4, 2 225 cm 400, 1. 2. 3. (a) (b) Square these numbers: 2, 4, 9, 11, 12, 18, Use your answers to (a) to find: 144 , 16, 121, 81, 324 Write down the following square roots without using a calculator: (a) (d) 9 169 (b) 36 225 (c) (D 100 Use a calculator to find these square roots, giving your answers correct to 2 decimal places: (a) (d) 6 20 (b) 10 50 43 (c) (D 12 90
  18. 2.5 4. 5. 6. 7. 8. 9. 10. MEP Y8 Practice Book A What are the lengths of the sides of a square which has an area of 81 cm 2 ? A square has an area of 140 cm . How long are the sides of this square, to the nearest mm? Explain why 51 < 8. Copy the statements below and complete each one, putting two consecutive whole numbers in the empty spaces: (a) (c) (b) (d) (f) Decide whether each of these statements is true or false: (a) 10 < 5 (c) 3.4< 12 < 3.5 (b) 2.6< 7

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