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Exercise-3.1

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Q1. Write all the factors of the following numbers :
(a) 24                (b) 15                (c) 21
(d) 27                (e) 12                (f) 20
(g) 18                (h) 23                (i) 36

Solution: Q1. 

(a) All factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

(b) All factors of 15 are 1, 3, 5, 15.

(c) All factors of 21 are 1, 3, 7, 21.

(d) All factors of 27 are 1, 3, 9, 27.

(e) All factors of 12 are 1, 2, 3, 4, 6, 12.

(f) All factors of 20 are 1, 2, 4, 5, 10, 20. 

(g) All factors of 18 are 1, 2, 3, 6, 9, 18.

(h) All factors of 23 are 1, 23.

(i) All factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Q2. Write first five multiples of :
(a) 5         (b) 8         (c) 9

Solutions: Q2. 

(a) First five multiples of 5 are 5, 10, 15, 20, 25. 

(b) First five multiples of 8 are 8, 16, 24, 32, 40.

(c) First five multiples of 9 are 9, 18, 27, 36, 45.

Q3. Match the items in column 1 with the items in column 2.

Column 1	Column 2
(i) 35	(a) Multiple of 8
(ii) 15	(b) Multiple of 7
(iii) 16	(c) Multiple of 70
(iv) 20	(d) Factor of 30
(v) 25	(e) Factor of 50
	(e) Factor of 20

Solution: Q3. 

(Answer is write in the front of column 1)

(i)35   (b) Multiple of 7

(ii)15   (d) Factor of 30

(iii)16   (a) Multiple of 8 

(iv)20   (f) Factor of 20

(v)25   (e) Factor of 50

Q4. Find all the multiples of 9 upto 100.

Solution: Q4. 

Answer- 9 x 1 = 9

        9 x 2 = 18

        9 x 3 = 27

        9 x 4 = 36

        9 x 5 = 45

        9 x 6 = 54 

        9 x 7 = 63

        9 x 8 = 72

        9 x 9 = 81 

        9 x 10 = 90

​        9 X 11 = 99 

Exercise-3.2 ch-3

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1. What is the sum of any two

(a) Odd numbers?          (b) Even numbers?
2. State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.

3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.

4. Write down separately the prime and composite numbers less than 20.

5. What is the greatest prime number between 1 and 10?

6. Express the following as the sum of two odd primes.
(a) 44        (b) 36        (c) 24        (d) 18
7. Give three pairs of prime numbers whose difference is 2.

[Remark : Two prime numbers whose difference is 2 are called twin primes].

8. Which of the following numbers are prime?
(a) 23     (b) 51      (c) 37      (d) 26
9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
10. Express each of the following numbers as the sum of three odd primes:
(a) 21     (b) 31          (c) 53         (d) 61
11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
(Hint : 3+7 = 10)
12. 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 ______.

Exercise 3.3

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1. Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572 (b) 726352 (c) 5500 (d) 6000 (e) 12159
(f) 14560 (g) 21084 (h) 31795072 (i) 1700 (j) 2150
2. Using divisibility tests, determine which of following numbers are divisible by 6:
(a) 297144 (b) 1258 (c) 4335 (d) 61233 (e) 901352
(f) 438750 (g) 1790184 (h) 12583 (i) 639210 (j) 17852
3. Using divisibility tests, determine which of the following numbers are divisible by 11:
(a) 5445              (b) 10824                    (c) 7138965      (d) 70169308        (e) 10000001                 (f) 901153

4. 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

Exercise 3.4 

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
6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
7. A number is divisible by 12. By what other numbers will that number be divisible?

Exercise 3.5 

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1. Which of the following statements are true?

(a) If a number is divisible by 3, it must be divisible by 9.      

Answer : False

(b) If a number is divisible by 9, it must be divisible by 3.       

Answer : True

(c) A number is divisible by 18, if it is divisible by both 3 and 6. 

Answer : False

(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.     

Answer : True

(e) If two numbers are co-primes, at least one of them must be prime.      

Answer : False

(f) All numbers which are divisible by 4 must also be divisible by 8. 

Answer : False

(g) All numbers which are divisible by 8 must also be divisible by 4. 

Answer : True

(h) If a number exactly divides two numbers separately, it must exactly divide their sum. 

Answer : True

(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately. 

Answer : False

2. Here are two different factor trees for 60. Write the missing numbers.

Solution:

There are two different way as follow: 

3. Which factors are not included in the prime factorization of a composite number?

Solution: 1 and the composite number itself not included in the prime factorization of a composite number.  

4. Write the greatest 4-digit number and express it in terms of its prime factors.

Solution: The greatest 4-digit number -

5. Write the smallest 5-digit number and express it in the form of its prime factors.

Solution: 

The smallest five diigit number is 10000.

It's tree factor is : 

Hence the prime factorisation =

2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 

6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

Solution:

Prime factors of 1729 are 7 × 13 × 19. 

7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

Solution:  Among the three consecutive numbers, there must be one even number and one multiple of 3. Thus, the product must be multiple of 6.

Example:

(i) 2 × 3 × 4 = 24

(ii) 4 × 5 × 6 = 120

8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

Solution: 3 + 5 = 8 and 8 is divisible by 4. 

5 + 7 = 12 and 12 is divisible by 4. 

7 + 9 = 16 and 16 is divisible by 4. 

9 + 11 = 20 and 20 is divisible by 4.

9. In which of the following expressions, prime factorisation has been done?

Solution:  In expressions (b) and (c), prime factorization has been done.

10. Determine if 25110 is divisible by 45.

[Hint: 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].

Solution: The prime factorization of 45 = 5 × 9 25110 is divisible by 5 as ‘0’ is at its unit place.

25110 is divisible by 9 as sum of digits is divisible by 9.

Therefore, the number must be divisible by 5 × 9 = 45

11. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.

Solution: No. Number 12 is divisible by both 6 and 4 but 12 is not divisible by 24.

12. I am the smallest number, having four different prime factors. Can you find me?

Solution: The smallest four prime numbers are 2, 3, 5 and 7.

Hence, the required number is 2 × 3 × 5 × 7 = 210

Exercise -3.6

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Q1. Find the H.C.F. of the following numbers:

(f) 34, 102

Solutions:

Prime factorization of 34

= 2 × 17

Prime factorization of 102

= 2 × 3 × 17

HCF of 34 and 102 is 2 × 17

= 34

(g) 70, 105, 175

Solution:

Prime factorization of 70

= 2 × 5 × 7

Prime factorization of 105

= 3 × 5 × 7

Prime factorization of 175

= 5 × 5 × 7

H.C.F. (70, 105, 175)

= 5 × 7

= 35

(h) 91, 112, 49

Solution:

Prime factorization of 91

= 7 × 13

Prime factorization of 112

= 2 × 2 × 2 × 2 × 7

Prime factorization of 49

= 7 × 7

H.C.F. (91, 112, 49) = 7

(i) 18, 54, 81

Solution:

Prime factorization of 18

= 2 × 3 × 3

Prime factorization of 54

= 2 × 3 × 3 × 3

Prime factorization of 81

= 3 × 3 × 3 × 3

H.C.F. = 3 × 3 = 9

(j) 12, 45, 75

Solution:

Prime factorization of 12

= 2 × 2 × 3

Prime factorization of 45

= 3 × 3 × 5

Prime factorization of 75

= 3 x 5 x 5

H.C.F. = 1 × 3

= 3

Q2. What is the H.C.F. of two consecutive:

(a) numbers?

(b) even numbers?

(c) odd numbers?

Solution:

(a) H.C.F. of two consecutive numbers be 1.

(b) H.C.F. of two consecutive even numbers be 2.

(c) H.C.F. of two consecutive odd numbers be 1.

Q3. H.C.F. of co-prime numbers 4 and 15 was found as follows by factorization:

4 = 2 × 2 and

15 = 3 × 5 since there is no common prime factor, so H.C.F. of 4 and 15 is 0. Is the answer correct? If not, what is the correct H.C.F.?

Solution: No. The correct H.C.F. is 1.

Exercise: 3.7

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Q1. 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.

Solution:

Renu purchases two bags of fertiliser of weights : 75 kg and 69 kg

Therefore, finding prime factorisation of 75 kg and 69 by dividing method:

75 = 3 × 5 × 5 

69 = 3 × 23 

Therefore, HCF of (75, 69) = 3 

Hence, Maximum Value of weight = 3 kg Answer

Q2. 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?

Solution:

Q3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.

Solution: 

The measurement of longest tape = H.C.F. (825 cm, 675 cm, 450 cm)

Therefore, the longest tape is 75 cm.

Q4. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.

Solution: 

L.C.M. of 6, 8 and 12 

= 2 × 2 × 2 × 3 

= 24

The smallest 3-digit number = 100

To find the number, we have to divide 100 by 24 1

00 = 24 × 4 + 4

Therefore, the required number = 100 + (24 – 4)

= 120.

Q 5. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

Solution:

The greatest 3-digit number = 999

Now we find LCM of 8, 10 and 12

So, When we subtract 39 from 999 we get 960.

Which is exactly divisible by 8, 10 and 12.

Hence, The greatest 3-digit number is 960.

Q6. 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?

Solution:

They meet together after LCM (48, 72, 108) seconds.

Now finding LCM,

Therefore, LCM (48, 72, 108) = 432

So these three lights burn together after 432 seconds.

432 seconds = 7 minutes 12 second

The time will be 7 a.m. +  7 minutes 12 second

= 7 : 07 : 12 a.m.

7. 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.

Solution:

The maximum capacity of that container = HCF (403, 434, 465)

Therefore, 31 litres of container is exact number which is required to measure the all containers' diesel.

8. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.

Solution: 

The least number will be LCM(6, 15,18) + 5 

Therefore, LCM(6, 15, 18) 

LCM = 2 × 3 × 3 × 5

     = 90

Now, 90 + 5 = 95

Hence required number 95 which leaves remainder as 5 on dividing by 6, 15, 18.

9. Find the smallest 4-digit number which is divisible by 18, 24 and 32.

Solution:

LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3

     = 288

The smallest 4-digit number = 1000

Now, dividing 1000 by 288

Remainder is 136.

1000 needs more (288-136) = 152 to divide exactly by 288

Therefore, Smallest required number is 1000 + 152 = 1152

10. Find the LCM of the following numbers :

(a) 9 and 4

(b) 12 and 5

(c) 6 and 5

(d) 15 and 4

Solution: 

(a) 9 and 4

9 and 4 has no common factors

Therefore, LCM = 9 × 4

= 36

Solution:

(b) 12 and 5

12 and 5 has no common factors

Therefore, LCM of 12 and 5 = 12 × 5

= 60

Solution:

(c) 6 and 5

6 and 5 has no common factors

Therefore, LCM of 6 and 5 = 6 × 5

= 30

Solution:

(d) 15 and 4

15 and 4 has no common factors

Therefore, LCM of 15 and 4 = 15 × 4

= 60

Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each case?

Yes, the L.C.M. is equal to the product of two numbers in each case. And L.C.M. is also the multiple of 3 in each case. 

11. Find the LCM of the following numbers in which one number is the factor of the other.

(a) 5, 20

(b) 6, 18

(c) 12, 48

(d) 9, 45

Solution: 

(a) 5, 20

5 and 20 has common factor 5. Both are divisible by 5 and five is one of them. 

Then, then greatest number will be LCM

LCM = 5 × 2 × 2 

Therefore, LCM = 20 

Solution:

(b) 6, 18

6 and 18 has common factor 6. and 6 is one of them. 

Then the greatest number will be LCM. 

LCM = 2 × 3 × 3

Therefore, LCM = 18

Solution: 

(c) 12, 48

12 and 48 has common factor 12 and 12 is one of them. 

Then the greatest number will be LCM.

LCM = 2 × 2 × 2 × 3

Therefore, LCM = 24

Solution: 

(d) 9, 45

9 and 45 has common factor 9 and 45 is one of them. 

Then the greatest number will be LCM

LCM = 3 × 3 × 5

Therefore, LCM = 45

What do you observe in the results obtained?

We observe that both numbers belongs to same table and greatest number is the LCM.