WEEK 3 4 Common Multiples Multiples that are

WEEK – 3 &4 Common Multiples : Multiples that are common to two numbers are known as common multiples of those numbers. Let us understand with the help of an example. Consider two numbers : 30 and 45. Multiples of 30 and 45 are 30 = 30, 60, 90, 120, 150, 180, 210, 240, 270…. . 45 = 45, 90, 135, 190, 225, 270………. We see that 90 and 270 are first two common multiples of 30 and 45. Common Factors that are common to two or more numbers are known as their common factors.

Considering the same two numbers 30 and 45. Factors of 30 and 45 are – 30 = {1, 2, 3, 4, 5, 6, 12, 15, 30} 45 = {1, 3, 5, 9, 15, 45} What are the common factors that you can observe? 15, 5, 3 and 1 appears in both 30 and 45. Common factors of 30 and 45 are 1, 3, 5 and 15.

Watch the link given below and try few exercises based on it. https: //www. youtube. com/watch? v=t. Ocx. ZMyveq. Q 1. Find the common factors of: (a) 20 and 28 (b) 15 and 25 (c) 35 and 50 (d) 56 and 120 2. Find the common factors of: (a) 4, 8 and 12 (b) 5, 15 and 25

3. Find first three common multiples of: (a) 6 and 8 (b) 12 and 18 4. Write all the numbers less than 100 which are common multiples of 3 and 4. 5. Which of the following numbers are co-prime? (a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 68 (e) 216 and 215 (f) 81 and 16

Here two ways to solve factorisation one is tree factorisation method and the other one is by dividing. ] For Example: Find prime factorisation of 32. Solution: Prime factorisation of 32 = 2 × 2 × 2. = 2⁵. Watch the link given below and try few questions based on it. https: //www. youtube. com/watch? v=t. W 97 UU 01 Sh. Y

Worksheet on Methods of Prime Factorization Practice the questions given in the worksheet on methods of prime factorization. 1. Each of the following is the prime factorization of a certain number. Find the number. (i) 2 × 5 × 7 (ix) 3 × 5 × 7 (ii) 3 × 7 (iii) 2 × 7 × 13 (iv) 2 × 3 × 5 (v) 7 × 11 (vi) 5 × 7 (vii) 3 × 7 (viii) 2 × 3 × 7

2. Find the prime factors by factor method. (i) 30 (ii) 36 (iii) 5 (iv) 27 (v) 72 (vi) 56 (vii) 80 (viii) 96 3. Determine the prime factorization by any method you like. (i) 2 (iii) 50 (v) 74 (vii)88 (ix) 81 (xi)66 (ii) 86 (iv) 68 (vi)90 (viii)48 (x)42 (xii)45

4. Here are two different factor trees for 60. Write the missing numbers. (a) (b)

5. Write the greatest 4 -digit number and express it in terms of its prime factors. 6. Write the smallest 5 -digit number and express it in the form of its prime factors. 7. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors. 8. In which of the following expressions, prime factorisation has been done? (a) 24 = 2 × 3 × 4 (b) 56 = 7 × 2 × 2 (c) 70 = 2 × 5 × 7 (d) 54 = 2 × 3 × 9

Highest Common Factor Highest common factor (H. C. F) of two or more numbers is the greatest number which divides each of them exactly. Now we will learn about the method of finding highest common factor (H. C. F). Steps 1: Find all the factors of each given number Step 2: Find common factors of the given number. Step 3: The greatest of all the factors obtained in Step 2, is the required highest common factor (H. C. F).

For Example: 1. Find the highest common factor (H. C. F) of 6 and 9. Factors of 6 = 1, 2, 3 and 6. Factors of 9 = 1, 3 and 9. Therefore, common factor of 6 and 9 = 1 and 3. Highest common factor (H. C. F) of 6 and 9 = 3. Therefore, 3 is H. C. F. or G. C. D. greatest common divisor of 6 and 9. H. C. F. or G. C. D. of given numbers is the greatest number which divides all the numbers without leaving a remainder.

2. Find the highest common factor (H. C. F) of 14 and 18. Factors of 14 = 1, 2, 7 and 14. Factors of 18 = 1, 2, 3, 6, 9 and 18. Therefore, common factor of 14 and 18 = 1 and 2. Highest common factor (H. C. F) of 14 and 18 = 2. Watch the link given below and try few exercises based on it. https: //www. youtube. com/watch? v=BWHSDl. IWNm 4

Word problems on H. C. F. and L. C. M. Here we will get the idea how to solve the word problems on H. C. F and L. C. M. 1. Find the smallest number which on adding 19 to it is exactly divisible by 28, 36 and 45. First we find the least common multiple (L. C. M. ) of 28, 36 and 45. Therefore, least common multiple (L. C. M. ) of 28, 36 and 45 = 2 × 3 × 5 × 7 = 1260 Therefore, the required number = 1260 - 19 = 1241

2. Find the number which divides 167 and 95 leaving 5 as remainder. The number divides 167 and leaves 5 as remainder Therefore, the number divides 167 - 5 = 162 exactly The number also divides 95 leaving 5 as remainder Therefore, the number divides 95 - 5 = 90 exactly Now we have to find highest common factor (H. C. F. ) of 162 and 90 Highest common factor (H. C. F. ) of 90 and 162 = 18 Therefore, 18 is the required number.

3. Find the largest number that divides 92 and 74 leaving 2 as remainder. The number divides 92 and leaves 2 as remainder Therefore, the number divides 92 - 2 = 90 exactly The number also divides 74 leaving 2 as remainder Therefore, the number divides 74 - 2 = 72 exactly Now we have to find highest common factor (H. C. F. ) of 90 and 72

Highest common factor (H. C. F. ) of 90 and 72 = 18 Therefore, 18 is the required number.

Worksheet on H. C. F. and L. C. M. I. Find highest common factor of the following by complete factorisation: (i) 48, 56, 72 (ii) 198, 360 (iii) 102, 68, 136 (iv) 1024, 576 (v) 405, 783, 513 II. Find the H. C. F. by long division method: (i) 84, 144 (ii) 120, 168 (iii) 430, 516, 817 (iv) 632, 790, 869 (v) 291, 582, 776 (vi) 219, 1321, 2320, 8526

III. Find the HCF of the following numbers : (a) 18, 48 (b) 30, 42 (c) 18, 60 (d) 27, 63 (e) 36, 84 (f) 34, 102 (g) 70, 105, 175 (h) 91, 112, 49 (i) 18, 54, 81 (j) 12, 45, 75 IV. Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times. V. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?

VI. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure three dimensions of the room exactly. VII. Determine the smallest 3 -digit number which is exactly divisible by 6, 8 and 12. VIII. Determine the greatest 3 -digit number exactly divisible by 8, 10 and 12. IX. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a. m. , at what time will they change simultaneously again?

X. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times. XI. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case. XII. Find the smallest 4 -digit number which is divisible by 18, 24 and 32. XIII. Find the LCM of the following numbers: (a) 9 and 4 (b) 12 and 5 (c) 6 and 5 (d) 15 and 4 Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each case?

After completing word problems, try Test – 1 and Test - 2 based on this chapter. Test - 1 1. Write all factors of each of the following numbers: (i) 60 (ii) 76 (iii) 125 (iv) 729 2. Write first five multiples of each of the following numbers: (i) 25 (ii) 35 (iii) 45 (iv) 40

3. Express each of the following numbers as the sum of three odd prime numbers: (i) 31 (ii) 35 (iii) 49 4. A list consists of the following pairs of numbers: (i) 51, 53; 55, 57; 59, 61; 63, 65; 67, 69; 71, 73 Categorize them as pairs of (i) co-primes (ii) primes (iii) Composites 5. Write seven consecutive numbers less than 100 so that there is no prime number between them.

6. Determine prime factorization of each of the following numbers: (i) 216 (ii) 420 (iii) 468 (iv) 945 (v) 7325 (vi) 13915 7. Find the HCF of the following numbers using prime factorization method: (i) 144, 198 (ii) 81, 117 (iii) 84, 98 (iv) 225, 450

(v) 170, 238 (vi) 504, 980 (vii) 150, 140, 210 (viii) 84, 120, 138 (ix) 106, 159, 265 TEST - 2 1. Find the largest number which divides 615 and 963 leaving remainder 6 in each case. 2. Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

3. Determine the LCM of the numbers given below: (i) 48, 60 (ii) 42, 63 (iii) 18, 17 (iv) 15, 30, 90 (v) 56, 65, 85 (vi) 180, 384, 144 (vii) 108, 135, 162 (viii) 28, 36, 45, 60 4. For each of the following pairs of numbers, verify the property: Product of the number = Product of their HCF and LCM (i) 25, 65 (ii) 117, 221

(iii) 35, 40 (iv) 87, 145 (v) 490, 1155 Mark the correct alternative in each of the following: 5. Which of the following numbers is a perfect number? (a) 4 (b) 12 (c) 8 (d) 6

6. The number of primes between 90 and 100 is (a) 0 (b) 1 (c) 2 (d) 3 7. The least prime is (a) 1 (b) 2 (c) 3 (d) 5 8. Which of the following numbers is divisible by 6? (a) 7908432 (b) 68719402 (c) 45982014

9. Fill in the blanks: (a) A number which has only two factors is called a ______. (b) A number which has more than two factors is called a ______. (c) 1 is neither ______ nor ______. (d) The smallest prime number is ______. (e) The smallest composite number is _____. (f) The smallest even number is ______. 10. Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3: (a) __ 6724 (b) 4765 __ 2

11. . Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no):

12. State true (T) or false (F): (i) The sum of primes cannot be a prime. (ii) The product of primes cannot be a prime. (iii) An even number is composite. (iv) Two consecutive numbers cannot be both primes. (v) Odd numbers cannot be composite. (vi) Odd numbers cannot be written as sum of primes. (vii) A number and its successor are always co-primes. (viii) If a number is divisible by 3, it must be divisible by 9. (ix) If a number is divisible by 9, it must be divisible by 3. (x) If a number is divisible by 4, it must be divisible by 8.

(xi) If a number is divisible by 8, it must be divisible by 4. (xii) A number is divisible by 18, if it is divisible by both 3 and 6. (xiii) If a number is divisible by both 9 and 10, it must be divisible by 90. (xiv) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately. (xv) If a number divides three numbers exactly, it must divide their sum exactly. (xvi) If two numbers are co-prime, at least one of them must be a prime number. (xvii) The sum of two consecutive odd numbers is always divisible by 4.