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Name of Lecturer: Mr. J.Agius Course: FCES LESSON 7 Multiples and Common Multiples, HCF and LCM 7.1 Multiples and Common Multiples Multiples are just all the multiplication tables. Example: The first 10 multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. Write which of these numbers are: Multiples of 7 1 2 3 4 5 6 7 8 9 10 Multiples of 9 11 12 13 14 15 16 17 18 19 20 Multiples of 13 21 22 23 24 25 26 27 28 29 30 Common multiples of 3 and 4 31 32 33 34 35 36 37 38 39 40 Multiples of both 2 and 3 41 42 43 44 45 46 47 48 49 50 Multiples of both 4 and 5 Common multiples of 5 and 6 Common multiples of 3 and 5 Learning Outcome 1 – Numerical Calculations Not multiples of 2, 3 or 4 Common multiples of 4 and 8 Page 51 Name of Lecturer: Mr. J.Agius Course: FCES Mixed Exercise 24 9 18 36 50 11 48 16 25 13 20 21 1 Multiples of 2 2 Multiples of 3 3 Multiples of 4 4 Common multiples of 2 and 3 5 Common multiples of 3 and 4 6 Common multiples of 3 and 7 7 Common multiples of 2 and 5 8 Least common multiple of 3 and 4 9 Least common multiple of 2 and 5 Learning Outcome 1 – Numerical Calculations Page 52 Name of Lecturer: Mr. J.Agius Course: FCES Common Multiples Write a number that is a multiple of: 1 2, 3 and 13 2 2, 3 and 9 3 2, 9 and 18 4 2, 3, 4 and 5 5 6, 8 and 15 6 12, 15 and 20 7 4, 6 and 9 8 3, 4 and 18 9 2, 3 and 21 10 4, 5, 6 and 10 7.2 Highest Common Factor When you are asked to mention the Highest Common Factor of two numbers or more, first you have to name all the factors of each number, then, verify the common factors, and hence determine the highest common factor. This number, in general is used to reduce fractions to their lowest term, by using the highest common factor of the numerator and the denominator. Example: The highest common factor of 20 and 15 is: Factors of 20 = 1, 2, 4, 5, 10, 20 Factors of 15 = 1, 3, 5, 15 The common factors are 1 and 5. So the highest common factor is 5. Learning Outcome 1 – Numerical Calculations Page 53 Name of Lecturer: Mr. J.Agius Course: FCES 7.3 Lowest Common Multiple Here we use the same procedure used in the Highest Common Factor. The lowest common multiple is used to transform two different fractions with different denominators, and change them to have same denominator. With this way you can add and subtract fractions. Example: The lowest common multiple of 4 and 5 is: Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, … Multiples of 5 = 5, 10, 15, 20, 25, … Here the lowest common multiple of 4 and 5 is 20. Exercise 1: Write down the highest common factor of a) 16 and 26 Ans.________ b) 36 and 45 Ans.________ c) 25 and 50 Ans.________ d) 56 and 84 Ans.________ Exercise 2: a) 5 and 8 Ans.________ b) 9 and 5 Ans.________ c) 13 and 2 Ans.________ d) 9 and 12 Ans.________ e) 4, 5 and 8 Ans.________ f) 8, 6 and 12 Ans.________ Find the lowest common denominator for each pair of fractions. 1. 2. Learning Outcome 1 – Numerical Calculations 3. Page 54 Name of Lecturer: Mr. J.Agius Course: FCES Story Sums 1. Look at these gears. A B a) A completes 10 turns. How many turns does B complete? b) What is the least number of turns that A can complete so that B also completes an exact number of turns? c) B completes 30 turns. On how many occasions will both A and B have been back in their starting position at the same time? 2. Graham’s three children, Angela, Bob and Cathy visit him on Sunday afternoons. Angela visits every 2 weeks, Bob visits every 3 weeks and Cathy visits every 4 weeks. All three children visited Graham last Sunday. How often do a) Angela and Bob visit on the same day? b) Angela and Cathy visit on the same day? c) Bob and Cathy visit on the same day? Learning Outcome 1 – Numerical Calculations Page 55 Name of Lecturer: Mr. J.Agius Course: FCES 3. James wants to draw the shapes below. a) In the list below, fill in the missing cells accordingly: i) ii) Find the Least Common Multiple (LCM) of the number of lines obtained for shapes D and F. Find the Highest Common Factor (HCF) of the number of lines obtained for shapes B and G. b) From the list above: i) arrange the numbers in order of size, smallest first; ii) write a prime number; iii) write down a multiple of 3; iv) write down a factor of 36. c) Write the total number of lines as a product of its prime factors in index form. (hint: 9 = 3 × 3 = 32 ) Learning Outcome 1 – Numerical Calculations Page 56 Name of Lecturer: Mr. J.Agius 4. Course: FCES A rectangular lawn measures 420cm by 240cm. What is the largest sized square tile that can be used to pave the area without any cutting? A box measures 12cm by 18cm by 24cm. The box is to be filled with cubes so that no 5. space is left. What is the length of the side of the largest cube that will do this? Learning Outcome 1 – Numerical Calculations Page 57 Name of Lecturer: Mr. J.Agius 6. Course: FCES Three bells start ringing together. If Bell A rings every 6 minutes, Bell B rings every 15 minutes and Bell C rings every 20 minutes. a) After how many minutes will Bell A and Bell B ring together? b) After how many minutes will all bells ring together? c) Think of another situation where such method can be used. Learning Outcome 1 – Numerical Calculations Page 58