NUMBER AND ALGEBRA Exercise 3.3 Prime numbers and composite numbers INDIVIDUAL PATHWAYS REFLECTION What strategies will you use to help recall the difference between prime and composite numbers? ⬛ ⬛ PRACTISE Questions: 1, 2a–f, 3a–f, 4a–f, 5a–f, 6a–d, 7 ⬛ CONSOLIDATE Questions: 1, 2d–i, 3d–i, 4d–i, 5d–i, 6c–f, 7 ⬛ ⬛ ⬛ Individual pathway interactivity MASTER Questions: 1, 2g–l, 3g–l, 4g–l, 5g–l, 6e–h, 7, 8 int-4320 FLUENCY 1 WE4 Find four prime numbers that are between 20 and 40. 2 WE5 State whether each of the following numbers is prime or composite. a 9 b 13 c 27 d 55 e 41 f 64 g 49 h 93 i 51 j 79 k 243 l 101 3 WE6 i Find the prime factors of each of the following numbers by drawing a factor tree. ii Write each one as a product of its prime factors in index form. a 15 b 30 c 24 d 100 e 49 f 72 g 18 h 56 i 45 j 84 k 98 l 112 WE7 4 i Find the prime factors of the following numbers by drawing a factor tree. ii Express the number as a product of its prime factors in index form. a 40 b 35 c 32 d 121 e 110 f 150 g 3000 h 64 i 96 j 196 k 90 l 75 5 Find the prime factors of each of the following numbers. a 48 b 200 c 42 d 81 e 18 f 39 g 27 h 300 i 60 j 120 k 50 l 80 6 WE8 Write the following as a product of prime factors using index notation. a 60 b 50 c 75 d 220 e 192 f 72 g 124 h 200 UNDERSTANDING 7 WE9 By expressing the following pairs of numbers as products of their prime factors, determine their lowest common multiple and their highest common factor. a 36 and 84 b 48 and 60 c 120 and 400 d 220 and 800 8 Can you find four prime numbers that are even? Explain. 9 Answer True (T) or False (F) for each of the following. a All odd numbers are prime numbers. b No even numbers are prime numbers. c 1, 2, 3 and 5 are the first four prime numbers. d A prime number has two factors only. 72 Maths Quest 7 NUMBER AND ALGEBRA 2 is the only even prime number The sum of two prime numbers is always even. The product of two prime numbers is always odd. There are no consecutive prime numbers. MC a The number of primes less than 10 is: A 4 B 3 C 5 D 2 E 1 b The first three prime numbers are: A 1, 3, 5 B 2, 3, 4 C 2, 3, 5 D 3, 5, 7 E 2, 5, 7 c The number 15 can be written as the sum of two prime numbers. These are: A 3 + 12 B 1 + 14 C 13 + 2 D 7+8 E 9+6 d Factors of 12 that are prime numbers are: A 1, 2, 3, 4 B 2, 3, 6 C 2, 3 D 2, 4, 6, 12 E 1, 2, 3, 4, 6, 12 Twin primes are pairs of primes that are separated from each other by one even number. For example, 3 and 5 are twin primes. Find two more pairs of twin primes. a Which of the numbers 2, 3, 4, 5, 6 and 7 cannot be the difference between two consecutive prime numbers? Explain. b For each of the numbers that can be a difference between two consecutive primes, give an example of a pair of primes less than 100 with such a difference. The following numbers are not primes. Each of them is the product of two primes. Find the two primes in each case. a 365 b 187 An easy way to find prime numbers is to use the Sieve of Eratosthenes. Eratosthenes discovered a simple method of sifting out all of the composite numbers so that only prime numbers are left. Alternatively, you can use the Excel file on your eBookPLUS. You can follow the following steps to find all prime numbers between 1 and 100. a Write the numbers from 1 to 100 in a grid as shown. e f g h 10 11 12 13 14 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 Digital doc Spreadsheet Sieve of Eratosthenes doc-1689 10 20 30 40 50 60 70 80 90 100 Cross out 1 as shown. It is not a prime number. Circle the first prime number, 2. Then cross out all of the multiples of 2. Circle the next prime number, 3. Now cross out all of the multiples of 3 that have not already been crossed out. e The number 4 is already crossed out. Circle the next prime number, 5. Cross out all of the multiples of 5 that are not already crossed out. b c d 5PQJDt Indices and primes 73 NUMBER AND ALGEBRA The next number that is not crossed out is 7. Circle 7 and cross out all of the multiples of 7 that are not already crossed out. g Do you need to check the multiples of any primes greater than 7? Why or why not? 15 MC a A factor tree for 21 is: f 21 A 7 3 1 b 3 1 7×1 21 1 7 3×1 21 E 3 21 C 1 21 D 21 B 21 3 7 7 A factor tree for 36 is: A B 36 2 18 9 36 D E 9 3 4 3 2 4 36 2 18 36 2 18 2 The prime factors of 16 are: A 1, 2 B 1, 2, 4 d The prime factors of 28 are: A 1, 28 B 2, 7 C 36 2 9 c C 2 D 1, 2, 4, 8, 16 E 2, 4, 8 C 1, 2, 14 D 1, 2, 7 E 2, 7, 14 REASONING 16 What is the largest three‐digit prime number in which each digit is a prime number? Prove that this number is a prime number. 17 Find a prime number greater than 10 where the sum of the digits equals 11. Show your working. 18 My age is a prime number. I am older than 50. The sum of the digits in my age is also a prime number. If you add a multiple of 13 to my age the result is 100. How old am I? PROBLEM SOLVING 19 Twin primes are pairs Digital doc WorkSHEET 3.1 doc-1687 74 Maths Quest 7 of prime numbers that differ by 2. Except for the pair of primes 2 and 3, this is the smallest difference between two prime numbers. The first twin primes are 3 and 5, followed by 5 and 7, then 11 and 13. What other twin primes are there below 100? 20 Find two prime numbers with a product of: a 21 b 26 c 323.

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