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Integers Objectives: C Grade B Grade Recognise Prime Numbers Find the Least Common Multiple (LCM) of two simple numbers Find the Highest Common Factor (HCF) of two simple numbers Write down a number as a product of Prime Factors Find the Least Common Multiple (LCM) of two or more numbers Find the Highest Common Factor (HCF) of two or more numbers Starter: Define: •an Integer •a Factor •a Multiple Integers any whole number positive or negative A number that divides exactly into another number (no remainder), e.g. 1 x 6 = 6, 2 x 3 = 6 so 1,2,3,6, are factors of 6, also x and y are factors of xy A multiple is in the times table of that number e.g. 3,6,9,12,15 are multiples of 3 x, 2x, 3x, 4x are multiples of x •a Prime Number has only 2 factors 1 and itself The first prime numbers are 2,3,5,7,11,13,17,19,23 •a Square Number The product of multiplying a number by itself e.g. 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16 So 1, 4, 9, 16 are square numbers The product of multiplying a number by itself three times 1, 8, 27, 64 are cube numbers •a Cube Number Integers How to find the factors of a number List all the ways you can make the number by multiplying other numbers: Find the factors of 36 1 x 36 = 36 Always start with 1 x the number 2 x 18 = 36 3 x 12 = 36 4 x 9 = 36 6 x 6 = 36 So 1,2,3,4,6,9,12,18 and 36 are factors of 36 Notice how the multiplications are In order of magnitude, you can use the tests of divisibility in a logical order to find factors Integers How to find the multiple of a number List all the times table of the number in ascending order. For bigger numbers add the number on each time. Find the multiples of 31 1 x 31 = 31 2 x 31 = 62 3 x 31 = 93 4 x 31 = 124 5 x 31 = 155 So the first 5 multiples of 31 are: 31, 62, 93, 124, 155 Integers Answer these: 1. Write down the factors of: 1, 3, 5, 15 a) 15 1, 2, 4, 8, 16, 32, 64 b) 64 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 c) 48 d) 32 1, 2, 4, 8, 16, 32 e) 72 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 f) 84 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 2. Write down the first 5 multiples of 3, 6, 9, 12, 15 a) 3 2, 4, 6, 8, 10 b) 2 5, 10, 15, 20, 25 c) 5 13, 26, 39, 52, 65 d) 13 15, 30, 45, 60, 75 e) 15 f) 16 16, 32, 48, 64, 80 Lowest Common Multiple (LCM) To find the lowest common multiple of two numbers, simply list the multiple of each number and the first number that appears in both lists is the LCM. e.g. Find the LCM of 8 and 14 8 16 24 32 40 48 56 64 72 80 14 28 42 56 70 84 56 is the first number to appear in each list and so 56 is the LCM Highest Common Factor (HCF) To find the Highest Common Factor of two numbers, simply list all the factors of each number and the largest number that appears in both lists is the HCF. Find the HCF of 18 and 42 18 1 2 3 6 9 18 42 1 2 3 6 7 14 21 42 6 is the largest number that appears in both groups therefore 6 is the HCF LCM and HCF Questions Answer these: 1. Write down the LCM of these sets of numbers: 30 a) 6 and 15 24 b) 3 and 8 60 c) 4, 10, 12 30 d) 3, 5, 6 96 e) 6, 8, 32 2. Write down the HCF of these pairs of numbers 3 a) 6 and 15 3 b) 12 and 15 16 c) 32 and 48 8 d) 56 and 152 7 e) 84 and 70 f) 27 and 36 9 Remember to use tests of divisibility if You are unsure of what the factors are LCM and HCF Harder Questions A lighthouse flashes every 56 seconds. Another lighthouse flashes every 40 seconds. At 9 p.m. they both flash at the same time. What time will it be when they next both flash at the same time? LCM question: List the multiples of both 56 112 168 224 280 336 40 80 120 160 200 240 280 320 280 is the LCM, so 280 seconds after 9 p.m. is when they next flash together. 280 60 = 4 minutes, 40 seconds 4 minutes and 40 seconds past 9 p.m. 9:00:00 + 0:04:40 = 9:04:40 LCM and HCF Harder Questions A farmer has 24 cows and 30 sheep. She decides to divide these equally between her sons. What is the greatest number of sons she can have? HCF question: List the factors of both: 24 1 2 3 4 6 8 12 24 30 1 2 3 5 6 10 15 30 6 is the HCF, so the maximum Number of sons she can have is 6