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Multiples, factors and primes. Contents. Multiples and factors. A. Prime numbers. A. A. Prime factor decomposition. HCF and LCM. A. MULTIPLES and FACTORS. Multiples. What are the first six multiples of 4?.

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  1. Multiples, factors and primes Contents Multiples and factors • A Prime numbers • A • A Prime factor decomposition HCF and LCM • A

  2. MULTIPLES and FACTORS

  3. Multiples What are the first six multiples of 4? A multiple of a number is found by multiplying the number by any whole number. To find the first six multiples of 4 multiply 4 by 1, 2, 3, 4, 5 and 6 in turn to get: 4, 8, 12, 16, 20 and 24. Any given number has infinitely many multiples.

  4. Multiples patterns on a hundred square

  5. Rectangular arrangements

  6. Finding factors A factor is a whole number that divides exactly into a given number. Factors come in pairs. For example, what are the factors of 30? 1 and 30, 2 and 15, 3 and 10, 5 and 6. So, in order, the factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.

  7. Factor finder

  8. Circle and square puzzle

  9. PRIME NUMBERS

  10. Sieve of Eratosthenes

  11. Prime numbers If a whole number has two, and only two, factors it is called a prime number. For example, the number 17 has only two factors, 1 and 17. Therefore, 17 is a prime number. The number 1 has only one factor, 1. Therefore, 1 is not a prime number. There is only one even prime number. What is it? 2 is the only even prime number.

  12. Prime numbers The first 10 prime numbers are: 3 5 7 11 13 17 19 23 29 2

  13. Testing for prime numbers Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive numbers. Is 107 divisible by 2? The last digit is a 7 so, no. Is 107 divisible by 3? The digit sum is 8 so, no. We don’t need to check for divisibility by 4 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it. Is 107 divisible by 5? The last digit is a 7 so, no. We don’t need to check for divisibility by 6 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.

  14. Testing for prime numbers Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive numbers. Is 107 divisible by 7? Dividing by 7 leaves a remainder so no. We don’t need to check for divisibility by 8 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it. We don’t need to check for divisibility by 9 because if 3 doesn’t divide into 107, then no multiple of 3 can divide into it. We don’t need to check for divisibility by 10 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.

  15. Testing for prime numbers Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive prime numbers. Why don’t we need to check for divisibility by 11? We don’t need to check for divisibility by 11 because we have found that no number below 10 divides into 107. That means that any number that multiplied 11 would have to be bigger than 10. Since, 10 × 11 is bigger than 107 we can stop here. 107 is a prime number.

  16. Testing for prime numbers When we are testing whether or not a number is prime, we only have to test for divisibility by prime numbers. We don’t need to check for divisibility by any number bigger than the square root of the number. A number is prime if no prime number less than the square root of the number divides into it. Also, all prime numbers greater than 5 must end in a 1, 3, 7 or 9.

  17. An amazing fact

  18. PRIME FACTORS

  19. Prime factors A prime factor is a factor that is also a prime number. For example, What are the factors of 30? The factors of 30 are: 1 2 3 5 6 10 15 30 The prime factors of 30 are 2, 3, and 5.

  20. Products of prime factors 2 × 3 × 5 = 30 2 × 2 × 2 × 7 = 56 This can be written as 23× 7 = 56 3 × 3 × 11 = 99 This can be written as 32× 11 = 99 Every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers.

  21. The prime factor decomposition When we write a number as a product of prime factors it is called the prime factor decomposition. For example, The prime factor decomposition of 100 is: 100 = 2 × 2 × 5 × 5 = 22× 52 There are 2 methods of finding the prime factor decomposition of a number.

  22. Factor trees 36 4 9 2 2 3 3 36 = 2 × 2 × 3 × 3 = 22× 32

  23. Factor trees 36 3 12 4 3 2 2 36 = 2 × 2 × 3 × 3 = 22× 32

  24. Factor trees 2100 30 70 6 5 10 7 2 3 2 5 2100 = 2 × 2 × 3 × 5 × 5 × 7 = 22× 3 × 52 × 7

  25. Factor trees 780 78 10 2 39 5 2 3 13 780 = 2 × 2 × 3 × 5 × 13 = 22× 3 × 5× 13

  26. Dividing by prime numbers 2 2 2 2 2 3 2 96 2 48 96 = 2 × 2 × 2 × 2 × 2 × 3 2 24 2 12 = 25 × 3 2 6 3 3 1

  27. Dividing by prime numbers 3 3 5 7 3 315 315 = 3 × 3 × 5 × 7 3 105 5 35 = 32 × 5 × 7 7 7 1

  28. Dividing by prime numbers 2 3 3 3 13 2 702 3 351 702 = 2 × 3 × 3 × 3 × 13 3 117 3 39 = 2 × 33× 13 13 13 1

  29. COMMON MULTIPLES HCF LCM

  30. Common multiples Multiples of 6 Multiples of 8 12 60 6 18 54 66 102… 24 48 72 96 8 16 24 32 40 48 56 64 72 80 88 96 104 … 30 42 78 90 84 36

  31. Multiples on a hundred grid

  32. The lowest common multiple The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers. We can find this by writing down the first few multiples for both numbers until we find a number that is in both lists. For example, 100, Multiples of 20 are : 20, 40, 60, 80, 120, . . . Multiples of 25 are : 25, 50, 75, 100, 125, . . . The LCM of 20 and 25 is 100.

  33. The lowest common multiple What is the lowest common multiple (LCM) of 8 and 10? The first ten multiples of 8 are: 8 16 24 32 40 48 56 64 72 80 … The first ten multiples of 10 are: 10 20 30 40 50 60 70 80 90 100 … The lowest common multiple (LCM) of 8 and 10 is 40.

  34. The lowest common multiple 5 5 and . Add together 4 4 12 12 9 9 × 4 × 3 31 + = + 36 36 36 × 4 × 3 We use the lowest common multiplewhen adding and subtracting fractions. For example, The LCM of 9 and 12 is 36. 16 15 =

  35. Common factor diagram

  36. The highest common factor The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers. We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists. For example, Factors of 36 are : 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 45 are : 1, 3, 5, 9, 15, 45. The HCF of 36 and 45 is 9.

  37. The highest common factor What is the highest common factor (HCF) of 24 and 30? The factors of 24 are: 1 2 3 4 6 8 12 24 The factors of 30 are: 1 2 3 5 6 10 15 30 The highest common factor (HCF) of 24 and 30 is 6.

  38. The highest common factor Cancel the fraction . 36 36 48 48 ÷12 ÷12 We use the highest common factor when cancelling fractions. For example, The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12. 3 = 4

  39. Using prime factors to find the HCF and LCM 1 1 We can use the prime factor decomposition to find the HCF and LCM of larger numbers. For example, Find the HCF and the LCM of 60 and 294. 2 60 2 294 2 30 3 147 3 15 7 49 5 5 7 7 60 = 2 × 2 × 3 × 5 294 = 2 × 3 × 7 × 7

  40. Using prime factors to find the HCF and LCM 60 294 60 = 2 × 2 × 3 × 5 294 = 2 × 3 × 7 × 7 2 7 2 3 5 7 HCF of 60 and 294 = 2 × 3 = 6 LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940

  41. Using prime factors to find the HCF and LCM

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