Chapter 3: Playing with numbers By Deepthi Menezes Chapter 3: Playing with numbers

Multiples Multiples of a number can be made by multiplying the number by any whole number. 1X12=12 2X12 = 24 3 X12 =36 4X12=48 12, 24, 36, 48, 60, 72, 84, 96,108,120….are multiples of 12        1 x 10 = 10,       2 x 10 = 20,       3 x 10 = 30,       4 x 10 = 40,       5 x 10 = 50,       6 x 10 = 60,       and so on ... Teacher Notes Ask the students to sing the table of 12. and as they sing write the multiples of 12 on the BB or display the ppt slide. Ask what do these numbers signify. Solicit that these are the product of two whole numbers. Multiples of a number can be made by multiplying the number by any whole number. The first 4 multiples of twelve are 12, 24, 36, 48 and you get the multiples by multiplying the 12 with 1, 2, 3, and 4. Similarly the numbers you find in the table of 10 are the multiples of 10. The first six multiples of 10 are 10, 20, 30, 40, 50 and 60

Multiples 5 10 15 20 25 30 35 40 45 50 55 A clock is set to ring at every 5th minute. Will it ring 55 minutes later? A clock is set to ring at every 5th minute. Will it ring on the 55 minutes later? Who thinks it will and who thinks it will not? Lets see it on a number line., by marking the chimes with dots on the number line. Agree that the watch will chime on the What do you notice about these marked numbers. That all these numbers are multiples of 5. Now lets see if 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. Give class work sheet and ask the students to do the sums. Assign 5 minutes for the written work. The rest they have to do for their homework.

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. Now lets see if 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. Give class work sheet and ask the students to do the sums. Assign 5 minutes for the written work. The rest they have to do for their homework.

What are the first five multiples of 13? 13 x 1 =13 13 x 2 = 26 13 x 3 = 39 13 x 4 = 52 13 x 5 = 65 13, 26, 39, 52, 65 13, 26, 39, 52, 65

Find the Missing Multiples 6, 12, 18, ____, ____ ___, 6, 9, 12, ____, ____, 21 ___, 24, 36, 48, 60, ____ 24 30 15 18 3 72 12 Class 5-Multiples &Factors, LCM &HCF

Composite and Prime numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Cross out 1, because it is not prime. Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number. Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2. Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number. Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100!

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.

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

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 In the given equation if 24 is the multip[le of 4 and 6 then what are 4 and 6 called factors. A factor of a number is an exact divisor of that number i.e. the factor leaves remainder 0 when it divides the number. 12 can be written as the product of 2 x 6 . If any number is divided by a number evenly (no remainder), then the second number is called a factor of the first number. For example, 6 is a factor of 12. Work sheet 12 is the multiple of its factors 2 and 6

FACTORS The factors of 12 are 1, 2, 3, 4, 6, and 12 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 Teacher notes: Sometimes a number can be divided evenly many ways: The factors of 12 are 1 x 12 2 x 6 3 x 4 4 x 3 6 x 2 12 x 2 In the given factor pairs 4x3 is the same as 3 x 4 and 6x 2 is the same as 2 x 6 , so while writing the all the factors of 12 , we don’t have to repeat these numbers. Thus the factor pairs if 12 are 1, 2, 3, 4, 6, and 12.

Factors For example, the product 24 has several factors. 24 = 1 x 24 So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 Teacher notes: Let us find the factors of 24. Class 5-Multiples &Factors, LCM &HCF

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. Teacher notes: To find the factors of a number we need to know the tests of divisibility. A number is divisible by 2  if the last digit is 0, 2, 4, 6 or 8. eg 168 is divisible by 2 since the last digit is 8. A number is divisible by 3  if the sum of the digits is divisible by 3. 168 is divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is divisible by 3 A number is divisible by 4  if the number formed by the last two digits is divisible by 4.eg 316 is divisible by 4 since 16 is divisible by 4. A number is divisible by 5  if the last digit is either 0 or 5.eg 195 is divisible by 5 since the last digit is 5. A number is divisible by 6  if it is divisible by 2 AND it is divisible by 3.eg 168 is divisible by 6 since it is divisible by 2 AND it is divisible by 3. A number is divisible by 8  if the number formed by the last three digits is divisible by 8.eg 7, 120 is divisible by 8 since 120 is divisible by 8. A number is divisible by 9  if the sum of the digits is divisible by 9.eg 549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9. A number is divisible by 10  if the last digit is 0.eg 1,470 is divisible by 10 since the last digit is 0.

1 x 48 2 x 24 3 x 16 4 x 12 6 x 8 Writing facors The factors of 48 are 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. Teacher notes: 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.

Factorization & Exponents 60=1x60  60=2x30  60=3x20  60=4x15  60=5x12  60=6x10 60 2 30 15 3 5

Factorization & Prime Factorization 60 The factor pairs for 60 are 60=1x60  60=2x30  60=3x20  60=4x15  60=5x12  60=6x10 Composite number 2 30 Composite number 3 10 Teacher Notes: What is factorization? Factors are the numbers you multiply together to get a product. To find the factors of 60 write all the factor pairs of 60. What is prime Factorization? "Prime Factorization" is finding which prime numbers you need to multiply together to get the original number. Now let us take a factor pair of 60 eg 2 x 30.2 is a prime number but 30 is a composite number as it is even and divisible. Next take 30 and factor it 3 times 10. 3 is a prime number but 10 is again a composite number. Factor 10 to 2 times 5. now both 2 and 5 are prime numbers. So the prime factors of 60 are 2x 2 x3 x5 Now lets see if we get the same answer if we take another factor pair. 5 2 60 is written as the product of its prime factors 60 = 2 x2 x 3 x 5

FACTOR TREE 60=4x15 =2x2 x3 x5 60 4 15 3 5 2 Even if we take another other set of factors of 60, on prime factorization we get the same set of prime numbers ie 2 x 2x 3x 5=60 Now lets try another method for finding the prime factors of 60. This method is through short division

Prime factorization through short division 60 30 15 2 3 5 1 The prime factors of 60 are 2x2x3x5 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. Teacher notes: What you do in this method is you start with the first prime, which is 2 and ask yourself whether the number is divisible by it. If it is, use short division to divide it by 2, and then continue with the answer you get from the division and ask yourself if that is still divisible by 2. If it is, divide that by 2. Eventually you will get an answer that is not divisible by 2, so then you ask yourself whether it is divisible by 3, and then continue on in this way with 5, 7, and the rest of the primes until you get an answer that is prime. Then the numbers along the side and the final answer will be the prime factorization written in the proper order.

Prime Factorization Now let us try another example: 910 So the prime factorization of 910 is 2x5x7x13

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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Teacher notes: Ask the students what do they understand by the term least common multiple. What are multiples: What is common? When we say common do we mean one thing or more than one? So what are we comparing here/ Numbers So what does the term least common multiple mean? A multiple which is the least and common to a set of numbers. Lets see the practical application of LCM in our daily life. 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?

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,... Teacher notes: 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. In the problem above, we have found multiples of the numbers 4 and 5. We have also found the Least Common Multiple (LCM) of 4 and 5, which is 20. Now lets do another sum.

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   Teacher notes Find the LCM of 18 and 24. What will you do first? Make a list of the multiples of 18 and 24. Identify the common multiples- 72 and 144 Which of these multiples is the least? 72 So the least common multiple of 18 and 24 is 72.

Highest Common Factor-HCF Find the HCF of 60 and 72 ,18 15 2 60 ,72 30 ,36 5 ,6 3 ,18 15 2 60 ,72 30 ,36 5 ,6 3 The common factors of 60 and 72 are 2 x 2 x 3 The HCF of 60 and 72 = 2x2x3=12 Uncommon factors

HCF by prime factorization Factors of 60 Factors of 72 Find the factors of 60 are Factors of 72 are The common factors are: The HCF of 60 and 72 is

LCM of 60 and 72 To find the LCM of 60 and 72 we multiply all the factors. 2x2 x3x5x6 = 336 So the HCF of 60 and 72 is 12 and LCM is 336 ,18 15 2 60 ,72 30 ,36 5 ,6 3

Find the HCF of 56 and 48 Prime Factor Tree for 56 Prime Factor Tree for 48 48 2 24 12 6 3 56 2 28 14 7 56 is all the prime numbers 2 x 2 x 2 x 7 multiplied together 48 is the prime numbers 2 x 2 x 2 x 2 x 3 multiplied together

48 2 24 12 6 3 56 28 14 7 8 x Finding the Highest Common Factor Look for common factors in both trees Multiply them together = 8 is the biggest number that goes into both 56 and 48 So it is the Highest Common Factor

HCF through Prime Factorization The prime factors of : 56 = 2 x 2 x 2 x 7 48 = 2 x 2 x 2 x 2 x 3 The prime numbers that are common are: 2x2x 2= 8 So the HCF of 56 and 48 is 8

HCF & LCM as a diagram 2 7 3 5 Write the common prime factors of both 504 and 700 here . The HCF =2X 2 X 7=28 Write the other PRIME FACTORS of 504 in the 504 circle Write the other PRIME FACTORS of 700 in the 700 circle 504 700 The LCM is found by multiplying all the numbers from the circles 504 = 2 x 2 x 2 x 3 x 3 x 7 700 = 2 x 2 x 5 x 5 x 7

The four different ways to find the HCF 504 2 7 3 5 700 ,18 15 2 60 ,72 30 ,36 5 ,6 3 x 48 2 24 12 6 3 56 28 14 7 8 The prime factors of : 56 = 2 x 2 x 2 x 7 48 = 2 x 2 x 2 x 2 x 3 The prime numbers that are common are: 2x2x 2= 8 So the HCF of 56 and 48 is 8