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© Boardworks Ltd 2005 1 of 64 N1 Integers KS4 Mathematics.

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1 © Boardworks Ltd 2005 1 of 64 N1 Integers KS4 Mathematics

2 © Boardworks Ltd 2005 2 of 64 Contents A A A A A N1.1 Classifying numbers N1 Integers N1.2 Calculating with integers N1.3 Multiples, factors and primes N1.4 Prime factor decomposition N1.5 LCM and HCF

3 © Boardworks Ltd 2005 3 of 64 Classifying numbers Natural numbers Positive whole numbers 0, 1, 2, 3, 4 … Integers Positive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, … Irrational numbers Numbers that cannot be expressed in the form n / m, where n and m are integers. Examples of irrational numbers are  and  2. Rational numbers Numbers that can be expressed in the form n / m, where n and m are integers. All fractions and all terminating and recurring decimals are rational numbers, for example, ¾, –0.63, 0.2..

4 © Boardworks Ltd 2005 4 of 64 Even numbers Even numbers are numbers that are exactly divisible by 2. All even numbers end in 0, 2, 4, 6 or 8. For example, 48 is an even number. It can be written as 48 = 2 × 24. E(3) = 6E(4) = 8E(5) = 10 Even numbers can be illustrated using dots or counters arranged as follows: E(1) = 2E(2) = 4 The n th even number can be written as E( n ) = 2 n.

5 © Boardworks Ltd 2005 5 of 64 Odd numbers Odd numbers leave a remainder of 1 when divided by 2. All odd numbers end in 1, 3, 5, 7 or 9. For example, 17 is an odd number. It can be written as 17 = 2 × 8 + 1 U(1) = 1 U(2) = 3U(3) = 5U(4) = 7U(5) = 9 Odd numbers can be illustrated using dots or counters arranged as follows: The n th odd number can be written as U( n ) = 2 n – 1.

6 © Boardworks Ltd 2005 6 of 64 Triangular numbers Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1. Triangular numbers can be illustrated using dots or counters arranged in triangles: For example, 15 is a triangular number. It can be written as 15 = 1 + 2 + 3 + 4 + 5 T(1) = 1 T(2) = 3 T(3) = 6 T(4) = 10 T(5) = 15

7 © Boardworks Ltd 2005 7 of 64 Triangular numbers Suppose we want to know the value of T(50), the 50 th triangular number. If we double the number of counters in each triangular arrangement we can make rectangular arrangements: We could either add together all the numbers from 1 to 50 or we could find a rule for the n th term, T( n ). T(1) = 1 T(2) = 3 T(3) = 6 T(5) = 15 2T(1) = 2 2T(2) = 6 2T(3) = 12 T(4) = 10 2T(4) = 20 2T(5) = 30

8 © Boardworks Ltd 2005 8 of 64 Triangular numbers Any rectangular arrangement of counters can be written as the product of two whole numbers: T(1) = 1 2T(1) = 2 T(2) = 3 2T(2) = 6 T(3) = 6 2T(3) = 12 T(4) = 10 2T(4) = 20 T(5) = 15 2T(5) = 30 2T(1) = 1 × 22T(2) = 2 × 32T(3) = 3 × 4 2T(4) = 4 × 52T(5) = 5 × 6 From these arrangements we can see that 2T( n ) = n ( n + 1) So, for any triangular number T( n ) T( n ) = n ( n + 1) 2

9 © Boardworks Ltd 2005 9 of 64 Triangular numbers We can now use this rule to find the value of the 50 th triangular number. T( n ) = n ( n + 1) 2 T(50) = 50(50 + 1) 2 T(50) = 50 × 51 2 T(50) = 2550 2 T(50) = 1275

10 © Boardworks Ltd 2005 10 of 64 Gauss’ method for adding consecutive numbers There is a story that when the famous mathematician Karl Friedrich Gauss was a young boy at school, his teacher asked the class to add up the numbers from one to a hundred. The teacher expected this activity to keep the class quiet for some time and so he was amazed when Gauss put up his hand and gave the answer, 5050, almost immediately!

11 © Boardworks Ltd 2005 11 of 64 Gauss’ method for adding consecutive numbers Gauss worked the answer out by noticing that you can quickly add together consecutive numbers by writing the numbers once in order and once in reverse order and adding them together. For example, to add the numbers from 1 to 10: 1+2+3+4+5+6+7+8+9+10 +9+8+7+6+5+4+3+2+1 11+ + + + + + + + + = 110 Sum of the numbers from 1 to 10 = 110 ÷ 2= 55 Use this method to show that the n th triangular number is: T( n ) = n ( n + 1) 2

12 © Boardworks Ltd 2005 12 of 64 Square numbers Square numbers are obtained when a whole number is multiplied by itself. They are sometimes called perfect squares. Square numbers can be illustrated using dots or counters arranged in squares: For example, 49 is a square number. It can be written as 49 = 7 × 7 or 49 = 7 2. S(1) = 1 S(2) = 4 S(3) = 9 S(4) = 16 S(5) = 25

13 © Boardworks Ltd 2005 13 of 64 Making square numbers The n th square number S( n ) can be written as S( n ) = n 2. We can multiply a whole number by itself. For example, 25 = 5 × 5 or 25 = 5 2. We can add consecutive odd numbers starting from 1. For example, 25 = 1 + 3 + 5 + 7 + 9. We can add together two consecutive triangular numbers. For example, 25 = 10 + 15 There are several ways to generate a sequence of square numbers. We can find the product of two consecutive even or odd numbers and add 1. For example, 25 = 4 × 6 + 1.

14 © Boardworks Ltd 2005 14 of 64 Difference between consecutive squares Show that the difference between two consecutive square numbers is always an odd number. If we use the general form for a square number n 2, where n is a whole number, we can write the square number following it as The difference between two consecutive square numbers can therefore be written as ( n + 1) 2 – n 2 =( n + 1)( n + 1) – n 2 = n 2 + n + n + 1 – n 2 = 2 n + 1 2 n + 1 is always an odd number for any whole number n. ( n + 1) 2.

15 © Boardworks Ltd 2005 15 of 64 Cube numbers Cube numbers are obtained when a whole number is multiplied by itself and then by itself again. Cube numbers can be illustrated using spheres arranged in cubes: For example, 64 is a cube number. It can be written as 64 = 4 × 4 × 4 or 64 = 4 3. C(2) = 8C(3) = 27C(4) = 64C(5) = 125 C(1) = 1 The n th cube number C( n ) can be written as C( n ) = n 3.

16 © Boardworks Ltd 2005 16 of 64 Squares, triangles and primes

17 © Boardworks Ltd 2005 17 of 64 Contents A A A A A N1.2 Calculating with integers N1 Integers N1.3 Multiples, factors and primes N1.4 Prime factor decomposition N1.5 LCM and HCF N1.1 Classifying numbers

18 © Boardworks Ltd 2005 18 of 64 Negative numbers A positive or negative whole number, including zero, is called an integer. For example, –3 is an integer. –3 is read as ‘negative three’. This can also be written as – 3. It is 3 less than 0. 0 – 3 =–3 Here the ‘–’ sign means minus 3 or subtract 3. We say, ‘zero minus three equals negative three’. Here the ‘–’ sign means negative 3.

19 © Boardworks Ltd 2005 19 of 64 Positive and negative integers can be shown on a number line. Positive integersNegative integers We can use the number line to compare integers. For example, –3–8 –3 > –8 –3 ‘is greater than’ –8 Integers on a number line

20 © Boardworks Ltd 2005 20 of 64 Adding integers We can use a number line to help us add positive and negative integers. –2 + 5 = -23 = 3 To add a positive integer we move forwards up the number line.

21 © Boardworks Ltd 2005 21 of 64 We can use a number line to help us add positive and negative integers. To add a negative integer we move backwards down the number line. –3 + –4 = = –7 -3-7 –3 + –4is the same as–3 – 4 Adding integers

22 © Boardworks Ltd 2005 22 of 64 5-3 Subtracting integers We can use a number line to help us subtract positive and negative integers. 5 – 8 == –3 To subtract a positive integer we move backwards down the number line. 5 – 8is the same as5 – +8

23 © Boardworks Ltd 2005 23 of 64 3 – –6 = 39 = 9 We can use a number line to help us subtract positive and negative integers. To subtract a negative integer we move forwards up the number line. 3 – –6is the same as3 + 6 Subtracting integers

24 © Boardworks Ltd 2005 24 of 64 We can use a number line to help us subtract positive and negative integers. –4 – –7 = -43 = 3 To subtract a negative integer we move forwards up the number line. –4 – –7is the same as–4 + 7 Subtracting integers

25 © Boardworks Ltd 2005 25 of 64 Adding and subtracting integers To add a positive integer we move forwards up the number line. To add a negative integer we move backwards down the number line. To subtract a positive integer we move backwards down the number line. To subtract a negative integer we move forwards up the number line. a + – b is the same as a – b. a – – b is the same as a + b.

26 © Boardworks Ltd 2005 26 of 64 Integer circle sums

27 © Boardworks Ltd 2005 27 of 64 When multiplying negative numbers remember: Rules for multiplying and dividing Dividing is the inverse operation to multiplying. When we are dividing negative numbers similar rules apply: +×+=+ –+×= – –+×=– –+×=– +÷+=+ –+÷= – –+÷=– –+÷=–

28 © Boardworks Ltd 2005 28 of 64 Multiplying and dividing integers Complete the following: –3 × 8 = 42 ÷ = –6 × –8 = 96 47 × = –141 –72 ÷ –6 = –36 ÷ = –4 ÷ –90 = –6 –7 × = 175 –4 × –5 × –8 = 3 × –8 ÷ = 1.5 –24 –7 –12 –3–3 12 9 540 –25 –160 –16

29 © Boardworks Ltd 2005 29 of 64 Using a calculator We can enter negative numbers into a calculator by using the sign change key: (–)(–) For example: –456 ÷ –6 can be entered as: (–)(–)456 ÷ (–)(–) 6 = The answer will be displayed as 76. Always make sure that answers given by a calculator are sensible.

30 © Boardworks Ltd 2005 30 of 64 What two integers have a sum of 2 and a product of –8? Sums and products Start by writing down all of the pairs of numbers that multiply together to make –8. Since –8 is negative, one of the numbers must be positive and one of the numbers must be negative. We can have: –1 × 8 = –81 × –8 = –8–2 × 4 = –8or 2 × –4 = –8 –1 + 8 = 71 + –8 = –7–2 + 4 = 22 + –4 = –2 The two integers are –2 and 4.

31 © Boardworks Ltd 2005 31 of 64 Sums and products

32 © Boardworks Ltd 2005 32 of 64 Contents A A A A A N1.3 Multiples, factors and primes N1 Integers N1.4 Prime factor decomposition N1.5 LCM and HCF N1.2 Calculating with integers N1.1 Classifying numbers

33 © Boardworks Ltd 2005 33 of 64 Multiples A multiple of a number is found by multiplying the number by any whole number. What are the first six multiples of 7? To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5 and 6 in turn to get: 7,14,21,28,35and 42. Any given number has infinitely many multiples.

34 © Boardworks Ltd 2005 34 of 64 Factors A factor (or divisor) of a number is a whole number that divides into it exactly. Factors come in pairs. For example, What are the factors of 30? 1 and30,2 and15,3 and10,5 and6. So, in order, the factors of 30 are: 1,2,3,5,6,10,15and 30.

35 © Boardworks Ltd 2005 35 of 64 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.

36 © Boardworks Ltd 2005 36 of 64 Prime numbers There are 25 prime numbers less than 100. These are: 2 13 31 53 73 3 17 37 59 79 5 19 41 61 83 7 23 43 67 89 11 29 47 71 97 What if we go above 100? Around 400 BC the Greek mathematician, Euclid, proved that there are infinitely many prime numbers.

37 © Boardworks Ltd 2005 37 of 64 Contents A A A A A N1.4 Prime factor decomposition N1 Integers N1.5 LCM and HCF N1.3 Multiples, factors and primes N1.2 Calculating with integers N1.1 Classifying numbers

38 © Boardworks Ltd 2005 38 of 64 A prime factor is a factor that is a prime number. For example, What are the prime factors of 70? The factors of 70 are: 12571014 3570 The prime factors of 70 are 2, 5, and 7. Prime factors

39 © Boardworks Ltd 2005 39 of 64 = 2 × 5 × 770 2 × 2 × 2 × 756 =This can be written as 56 = 2 3 × 7 = 3 × 3 × 1199This can be written as 99 = 3 2 × 11 Every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers. Products of prime factors

40 © Boardworks Ltd 2005 40 of 64 The prime factor decomposition When we write a number as a product of prime factors it is called the prime factor decomposition or prime factor form. For example, The prime factor decomposition of 100 is: There are two methods of finding the prime factor decomposition of a number. 100 = 2 × 2 × 5 × 5 = 2 2 × 5 2

41 © Boardworks Ltd 2005 41 of 64 36 49 2233 36 = 2 × 2 × 3 × 3 = 2 2 × 3 2 Factor trees

42 © Boardworks Ltd 2005 42 of 64 36 312 43 22 36 = 2 × 2 × 3 × 3 = 2 2 × 3 2 Factor trees

43 © Boardworks Ltd 2005 43 of 64 2100 3070 65 23 107 25 2100 = 2 × 2 × 3 × 5 × 5 × 7 = 2 2 × 3 × 5 2 × 7 Factor trees

44 © Boardworks Ltd 2005 44 of 64 780 7810 392 313 25 780 = 2 × 2 × 3 × 5 × 13 = 2 2 × 3 × 5 × 13 Factor trees

45 © Boardworks Ltd 2005 45 of 64 962 482 242 122 62 33 1 2 2 2 2 2 3 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2 5 × 3 Dividing by prime numbers

46 © Boardworks Ltd 2005 46 of 64 3153 1053 355 77 1 3 3 5 7315 = 3 × 3 × 5 × 7 = 3 2 × 5 × 7 Dividing by prime numbers

47 © Boardworks Ltd 2005 47 of 64 7022 3513 1173 393 13 1 2 3 3 3 702 = 2 × 3 × 3 × 3 × 13 = 2 × 3 3 × 13 Dividing by prime numbers

48 © Boardworks Ltd 2005 48 of 64 Using the prime factor decomposition Use the prime factor form of 324 to show that it is a square number. 3242 1622 813 273 93 33 1 2 2 3 3 3 3 324 = 2 × 2 × 3 × 3 × 3 × 3 = 2 2 × 3 4 This can be written as: (2 × 3 2 ) × (2 × 3 2 ) or (2 × 3 2 ) 2 If all the indices in the prime factor decomposition of a number are even, then the number is a square number.

49 © Boardworks Ltd 2005 49 of 64 Using the prime factor decomposition Use the prime factor form of 3375 to show that it is a cube number. 33753 11253 3753 1255 255 55 1 3 3 3 5 5 5 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3 3 × 5 3 This can be written as: (3 × 5) × (3 × 5) × (3 × 5) or (3 × 5) 3 If all the indices in the prime factor decomposition of a number are multiples of 3, then the number is a cube number.

50 © Boardworks Ltd 2005 50 of 64 Using the prime factor decomposition 168 = 2 3 × 3 × 7 4116 = 2 2 × 3 × 7 3 294 = 2 × 3 × 7 2 Use the prime factor decompositions of the numbers given above to answer the following questions. 1) What is 168 × 294 as a product of prime factors? 168 × 294 = (2 3 × 3 × 7) × (2 × 3 × 7 2 ) = 2 3 × 2 × 3 × 3 × 7 × 7 2 = 2 4 × 3 2 × 7 3

51 © Boardworks Ltd 2005 51 of 64 Using the prime factor decomposition 168 = 2 3 × 3 × 7 4116 = 2 2 × 3 × 7 3 294 = 2 × 3 × 7 2 Use the prime factor decompositions of the numbers given above to answer the following questions. 2) What is 4116 ÷ 294? 4116 ÷ 294 = 2 2 × 3 × 7 3 2 × 3 × 7 2 = 2 × 2 × 3 × 7 × 7 × 7 2 × 3 × 7 × 7 = 2 × 7 = 14

52 © Boardworks Ltd 2005 52 of 64 Using the prime factor decomposition 168 = 2 3 × 3 × 7 4116 = 2 2 × 3 × 7 3 294 = 2 × 3 × 7 2 Use the prime factor decompositions of the numbers given above to answer the following questions. 3) Is 4116 divisible by 168? If we divide 4116 by 168 we have: 4116 ÷ 168 = 2 2 × 3 × 7 3 2 3 × 3 × 7 = 2 × 2 × 3 × 7 × 7 × 7 2 × 2 × 2 × 3 × 7 There is a 2 left in the denominator No, 4116 is not divisible by 168.

53 © Boardworks Ltd 2005 53 of 64 Using the prime factor decomposition 168 = 2 3 × 3 × 7 4116 = 2 2 × 3 × 7 3 294 = 2 × 3 × 7 2 Use the prime factor decompositions of the numbers given above to answer the following questions. 4) Show that 294 × 6 is a square number. We can write 6 as 2 × 3 294 × 6 = 2 × 3 × 7 2 × 2 × 3 Rearranging, 294 × 6 = 2 × 2 × 3 × 3 × 7 2 = 2 2 × 3 2 × 7 2 = (2 × 3 × 7) 2

54 © Boardworks Ltd 2005 54 of 64 Using the prime factor decomposition 168 = 2 3 × 3 × 7 4116 = 2 2 × 3 × 7 3 294 = 2 × 3 × 7 2 Use the prime factor decompositions of the numbers given above to answer the following questions. 5) Write the fraction in its simplest form. 168 294 168 294 = 2 3 × 3 × 7 2 × 3 × 7 2 2 × 2 × 2 × 3 × 7 2 × 3 × 7 × 7 = = 4 7

55 © Boardworks Ltd 2005 55 of 64 Contents A A A A A N1.5 LCM and HCF N1 Integers N1.4 Prime factor decomposition N1.3 Multiples, factors and primes N1.2 Calculating with integers N1.1 Classifying numbers

56 © Boardworks Ltd 2005 56 of 64 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. For small 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, Multiples of 20 are :20,40,60,80, 100, 120,... Multiples of 25 are :25,50,75,100,125,... The LCM of 20 and 25 is100.

57 © Boardworks Ltd 2005 57 of 64 We use the lowest common multiple when adding and subtracting fractions. For example, Add together 4 9 5 12 and The LCM of 9 and 12 is 36. + 4 9 5 12 = 36 × 4 16 + 36 × 3 15 = 31 36 The lowest common multiple

58 © Boardworks Ltd 2005 58 of 64 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 is9.9.

59 © Boardworks Ltd 2005 59 of 64 We use the highest common factor when cancelling fractions. For example, Cancel the fraction 36 48 The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12. 36 48 = ÷12 3 4 The highest common factor

60 © Boardworks Ltd 2005 60 of 64 Using prime factors to find the HCF and LCM 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 125. 602 302 153 55 1 60 = 2 × 2 × 3 × 5 2942 1473 497 77 1 294 = 2 × 3 × 7 × 7

61 © Boardworks Ltd 2005 61 of 64 60 294 60 = 2 × 2 × 3 × 5 294 = 2 × 3 × 7 × 7 2 2 3 5 7 7 HCF of 60 and 294 =2 × 3= 6 LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 =2940 Using prime factors to find the HCF and LCM

62 © Boardworks Ltd 2005 62 of 64 Using prime factors to find the HCF and LCM

63 © Boardworks Ltd 2005 63 of 64 The LCM of co-prime numbers If two numbers have a highest common factor (or HCF) of 1 then they are called co-prime or relatively prime numbers. For two whole numbers a and b we can write: If two whole numbers a and b are co-prime then: For example, the numbers 8 and 9 do not share any common multiples other than 1. They are co-prime. Therefore,LCM(8, 9) = 8 × 9 =72 a and b are co-prime if HCF( a, b ) = 1 LCM( a, b ) = ab

64 © Boardworks Ltd 2005 64 of 64 The LCM of numbers that are not co-prime If two numbers are not co-prime then their highest common factor is greater than 1. If two numbers a and b are not co-prime then their lowest common multiple is equal to the product of the two numbers divided by their highest common factor. We can write this as: For example, LCM( a, b ) = ab HCF( a, b ) LCM(8, 12) = 8 × 12 HCF(8, 12) = 96 4 = 24


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