By Deepthi Menezes Chapter 3 Playing with numbers
By Deepthi Menezes Chapter 3: Playing with numbers 1
Multiples �Multiples of a number 1 x 10 = 10, can be made by 2 x 10 = 20, multiplying the number 3 x 10 = 30, by any whole number. 4 x 10 = 40, � 1 X 12=12 5 x 10 = 50, � 2 X 12 = 24 6 x 10 = 60, � 3 X 12 =36 and so on. . . � 4 X 12=48 12, 24, 36, 48, 60, 72, 84, 96, 108, 120…. are multiples of 12 2
Multiples � 5 10 15 20 25 30 35 40 45 50 55 A clock is set to ring at every 5 th minute. Will it ring 55 minutes later? 3
Multiples � Is 12 a multiple of 3? If you multiply 3 by 4 you get 12, so 12 is a multiple of 3. � Is 15 a multiple of 3? 3 x 5 = 15. So 15 is a multiple of 3, (and also of 5). � Is 21 a multiple of 6? 21 is not a multiple of 6 because you can't make 21 by multiplying 6 by any whole number. 6 x 3 = 18 and 6 x 4 = 24 but there is no whole number between 3 and 4 that could give us an answer of 21.
What are the first five multiples of 13? 13 x 1 =13 13 x 2 = 26 13 x 3 = 39 13, 26, 39, 52, 65 13 x 4 = 52 13 x 5 = 65 5
Find the Missing Multiples 24 30 � 6, 12, 18, ____ 15 18 3 �___, 6, 9, 12, ____, 21 �___, 24, 36, 48, 60, ____ 72 12 Class 5 -Multiples &Factors, LCM &HCF 6
Composite and Prime numbers 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 10 20 30 40 50 60 70 80 90 100 7
Prime Numbers � A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. � For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. � If we list the factors of 29, we only have 1 and 29. That's two factors. So we say that 29 is a prime number, but 28 isn't. 8
Composite Numbers �A Composite Number can be divided evenly by numbers other than 1 or itself. � Example: is 6 a Prime Number or Composite Number? � 6 can be divided evenly by 2, or by 3, as well as by 1 or 6: � 6=1× 6 6=2× 3 � So 6 is a Composite Number Class 5 -Multiples &Factors, LCM &HCF 9
Factors �Factors are the numbers you multiply together to get a product or a factor is a number that exactly divides another number without leaving a remainder. � 12 can be written as the product of 2 x 6 or 2 x 6 = 12 2 and 6 are the factors of 12 12 is the multiple of its factors 2 and 6 10
FACTORS The factors of 12 are: � 1 x 12 2 x 6 3 x 4 4 x 3 6 x 2 12 x 1 12 can be divided evenly by 1, 2, 3, 4, 6 and 12: 1 × 12 = 12 2 × 6 = 12 3 × 4 = 12 � So 12 is a Composite Number �The factors of 12 are 1, 2, 3, 4, 6, and 12 11
Factors �For example, the product 24 has several factors. � 24 = 1 x 24 � 24 = 2 x 12 � 24 = 3 x 8 � 24 = 4 x 6 �So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 Class 5 -Multiples &Factors, LCM &HCF 12
Divisibility rules � A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8. � A number is divisible by 3 if the sum of the digits is divisible by 3. � A number is divisible by 4 if the number formed by the last two digits is divisible by 4. � A number is divisible by 5 if the last digit is either 0 or 5. � A number is divisible by 10 if the last digit is 0. � A number is divisible by 8 if the number formed by the last three digits is divisible by 8. � A number is divisible by 9 if the sum of the digits is divisible by 9. � A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3. 13
Writing facors The factors of 48 are 1 x 48 2 x 24 3 x 16 4 x 12 6 x 8 Another way of writing factors is: Write your first pair of factors with a reasonable space between them, then move on to the next pair until you have them all. This way, when you get to the 6, 8 pair, you can stop because 7 is not a factor and you already have 8 in your list. 14
Factorization & Exponents 60 � 60=1 x 60 60=2 x 30 60=3 x 20 60=4 x 15 60=5 x 12 60=6 x 10 30 2 2 15 3 5 15
Factorization & Prime Factorization 60 �The factor pairs for 60 are � 60=1 x 60 60=2 x 30 60=3 x 20 60=4 x 15 60=5 x 12 60=6 x 10 Composite number 2 30 Composite number 3 10 2 5 60 is written as the product of its prime factors 60 = 2 x 3 x 5
FACTOR TREE � 60=4 x 15 60 =2 x 2 x 3 x 5 4 2 15 2 3 5
Prime factorization through short division 2 60 2 30 3 15 5 5 1 The prime factors of 60 are 2 x 2 x 3 x 5 2, 3, and 5 are all prime numbers, so we have prime factored 60. All we have to do now is neaten our answer up a bit. It is customary to write prime factorizations in increasing order, that is with the smallest numbers first.
Prime Factorization �Now let us try another example: 910 So the prime factorization of 910 is 2 x 5 x 7 x 13 19
Least common Multiple-LCM � A cold drink truck visits Rita's neighbourhood every 4 days and Amul ice cream truck visits her neighborhood every 5 days. For the month of June on which day will both the trucks visit on the same day? 1 8 15 22 29 2 9 16 23 30 3 10 17 24 4 11 18 25 5 12 19 26 6 13 20 27 7 14 21 28 20
Least common Multiple-LCM In the given problem we have to first find the multiples of both the numbers. � Cross out the multiples that are common. (20 and 40) � Which multiple is the least-20 � Both the trucks will visit the neighbour hood on 20 June or after 20 days. � Truck Days of visit 1 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, . . . 2 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, . . . 21
Exercise -Find the LCM of 18 and 24. Multiples of 18 18, 36, 54, 72, 90, 108, 126, 144, … Multiples of 24 24, 48, 72, 96, 120, 144, …. • • • Common multiples of 18 and 24 are 72 and 144 The least common multiple of 18 and 24 is 72. LCM = 72
Highest Common Factor-HCF Find the HCF of 60 and 72 2 60 , 72 2 30 , 36 3 15 , 18 5 , 6 The common factors of 60 and 72 are 2 x 3 The HCF of 60 and 72 = 2 x 2 x 3=12 2 60 , 72 2 30 , 36 3 15 , 18 5 , 6 Uncommon factors
HCF by prime factorization Factors of 60 Factors of 72
LCM of 60 and 72 �To find the LCM of 60 and 72 we multiply all the factors. � 2 x 2 x 3 x 5 x 6 = 336 �So the HCF of 60 and 72 is 12 and �LCM is 336 2 60 , 72 2 30 , 36 3 15 , 18 5 , 6
Find the HCF of 56 and 48 Prime Factor Tree for 56 � Prime Factor Tree for 48 48 56 2 28 2 2 14 2 24 2 7 56 is all the prime numbers 2 x 2 x 7 multiplied together 6 2 3 48 is the prime numbers 2 x 2 x 3 multiplied together
Finding the Highest Common Factor 48 56 2 28 2 x 14 2 2 12 2 2 7 24 2 x 2 2 Look for common factors in both trees Multiply them together = 8 6 2 8 is the biggest number that goes into both 56 and 48 So it is the Highest Common Factor 3
HCF through Prime Factorization �The prime factors of : 56 = 2 x 2 x 7 48 = 2 x 2 x 3 The prime numbers that are common are: 2 x 2 x 2= 8 So the HCF of 56 and 48 is 8
HCF & LCM as a diagram Write the common prime factors of both 504 and 700 here. The HCF =2 X 2 X 7=28 504 = 2 x 2 x 3 x 7 700 = 2 x 5 x 7 504 700 2 3 3 5 2 2 7 5 Write the other PRIME FACTORS of 700 in the 700 circle Write the other PRIME FACTORS of 504 in the 504 circle The LCM is found by multiplying all the numbers from the circles
The four different ways to find the HCF 504 700 2 3 3 5 2 2 7 2 30 , 36 5 The prime factors of : 56 = 2 x 2 x 7 48 = 2 x 2 x 3 The prime numbers that are common are: 2 x 2 x 2= 8 So the HCF of 56 and 48 is 8 60 , 72 2 15 , 18 5 , 6 3 48 56 2 28 2 x 2 14 2 2 2 7 24 12 2 x 2 2 8 6 2 3
- Slides: 30