Chapter 3: Playing with Numbers

Ex- 3.1

1. 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:

  (a) 1, 2, 3, 4, 6, 8, 12, 24

(b) 1, 3, 5, 15

 (c) 1, 3, 7, 21

 (d) 1, 3, 9, 27

 (e) 1, 2, 3, 4, 6, 12

 (f) 1, 2, 4, 5, 10, 20

 (g) 1, 2, 3, 6, 9, 18

 (h) 1, 23

 (i) 1, 2, 3, 4, 6, 9, 12, 18, 36

2. Write first five multiples of:

 (a) 5     (b) 8     (c) 9

Solution:

 (a) 5, 10, 15, 20, 25

 (b) 8, 16, 24, 32, 40

 (c) 9, 18, 27, 36, 45

3. 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

                                        (f) Factor of 20

Solution:

 (i) → (b)

 (ii) → (d)

 (iii) → (a)

 (iv) → (f)

 (v) → (e)

4. Find all the multiples of 9 up to 100.

Solution:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.

EXERCISE 3.2

1. What is the sum of any two (a) Odd numbers? (b) Even numbers?

Solution:

a. Odd number

1 + 1 = 2

1 + 3 = 4

5 + 7 = 12

9 + 11= 20

The sum of two odd number is always an even number.

b. Even number

2 + 2 = 4

4 + 4 = 8

2 + 6 = 8

8 + 8 = 16

The sum of any two even number is always an even number.

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.

Solution:

(a) F

(b) T

 (c) T

 (d) F

(e) F

(f) F

(g) F

 (h) T

 (i) F

(j) T

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

Solution:

17 and 71,

 37 and 73,

 79 and 97.

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

Solution:

01,02,03,04,05,06,07,08,09,10,11,12,13,14,15,16,17,18,19,20

Prime number are = 02,03,05,07,11,13,17,19.

Composite number are = 04,06,008,09,10,12,14,15,16,18,20.

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

Solution:

07 is the greatest prime number between 01 and 10.

6. Express the following as the sum of two odd primes.

 (a) 44 (b) 36 (c) 24 (d) 18

Solution:

(a) 3 + 41

(b) 5 + 31

 (c) 5 + 19

 (d) 5 + 13

(This could be one of the ways. There can be other ways also.)

7. Give three pairs of prime numbers whose difference is 2. [Remark: Two prime numbers whose difference is 2 are called twin primes].

Solution:

(3, 5), (5, 7), (11, 13)

8. Which of the following numbers are prime?

 (a) 23 (b) 51 (c) 37 (d) 26

Solution:

a.23

1*23 =23;

23 is prime number.

b. 51

1*51 = 51;

3*17 = 51;

51 is not a prime number.

It is a composite number.

c.37

1*37 = 37;

37 is prime number.

d.26

1*26 = 26;

2*13 = 26;

26 is not a prime number.

It is a composite number.

9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.

Solution:

Seven consecutive composite number are

90,91,92,93,94,95,96.

10. Express each of the following numbers as the sum of three odd primes:

  (a) 21 (b) 31 (c) 53 (d) 61

Solution:

 (a) 21

= 3 + 5 + 13

= 21

(b) 31

= 3 + 5 + 23

= 31

 (c) 53

= 13 + 17 + 23

= 53

 (d) 61

= 7 + 13 + 41

= 61

 (This could be one of the ways. There can be other ways also.)

11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.

 (Hint: 3+7 = 10)

Solution:

(2, 3);

(2, 13);

(3, 17);

(7, 13);

(11, 19);

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 ______.

Solution:

 (a) prime number

 (b) composite number

(c) prime number, composite number

 (d) 2

 (e) 4

 (f) 2

Ex- 3.3

1. 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):

Solution:

2. 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

Solution:

Divisible by 4 :

 (a) 572,

 (b) 726352,

 (c) 5500,

(d) 6000,

 (f) 14560,

 (g) 21084,

(h) 31795072,

 (I) 1700;

Divisible by 8:

 (b) 726352,

 (d) 6000,

 (f) 14560,

 (h) 31795072,

3. 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

Solution:

Divisible by 6:

(a) 297144,

 (f) 428750,

 (g) 1790184,

 (i) 639210;

4. 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

Solution:

Divisible by 11:

(a) 5445,

 (b)10824,

 (d) 70169308,

 (e) 10000001,

 (f) 901153;

5. 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

Solution:

(a) 2 and 8

(b) 0 and 9

6. Write a digit in the blank space of each of the following numbers so that the number formed is divisible by 11:

(a) 92 __ 389

 (b) 8 __ 9484

Solution:

 (a) 8

 (b) 6

EXERCISE 3.4

1. Find the common factors of:

(a) 20 and 28

(b) 15 and 25

 (c) 35 and 50

 (d) 56 and 12

Solution:

(a) 20 and 28

Factors of 20 are 1,2,4,5,20.

Factors of 28 are 1,2,4,7,28.

Common factors of 20 and 28

= 1, 2, 4

 (b) 15 and 25

Factors of 15 are 1,3,5,15.

Factors of 25 are 1,5,25.

Common factors of 15 and 25

= 1, 5

 (c) 35 and 50

Factors of 35 are 1,5,7,35.

Factors of 50 are 1,2,5,25,50.

Common factors of 35 and 50

= 1, 5

 (d) 56 and 120

Factors of 56 are 1,2,4,7,8,14,15.

Factors of 120 are 1,2,3,3,4,6,8,10,12,18,20.

Common factors of 56 and 12

= 1, 2, 4, 8

2. Find the common factors of:

(a) 4, 8 and 12

 (b) 5, 15 and 25

Solution:

(a) 4,8 and 12

Factors of 4 are 1,2,4.

Factors of 8 are 1,2,4,8.

Factors of 12 are 1,2,3,4,6,12.

Common factors of 4,8 and 12

= 1,2,4

(b) 5,15 and 25

Factors of 5 are 1,5.

Factors of 15 are 1,3,5,15.

Factors of 25 are 1,5,25.

Common factors of 5,15 and 25

= 1,5

3. Find first three common multiples of:

 (a) 6 and 8 (b) 12 and 18

Solution:

 (a) 6 and 8

Multiples of 6 are 6,12,18,24,30,36,42,48,54,60.

Multiples of 8 are 8,16,24,32,40,48,56,64,72,80.

Common multiples of 6 and 8

= 24, 48, 72

 (b) 12 and 18

Multiples of 12 are 12,24,36,48,60,72,84,96,108,120.

Multiples of 18 are 18,36,54,72,90,108,126,144,162,180.

Common multiples of 12 and 18

= 36, 72, 108

4. Write all the numbers less than 100 which are common multiples of 3 and 4.

Solution:

Multiples of 3 are 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81, 84,87,90,93,96,99.

Multiples of 4 are 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100.
Common multiples of 3 and 4 are

= 12, 24, 36, 48, 60, 72, 84, 96.

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

Solution:

(a)18 and 35,

 (b) 15 and 37,

 (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?

Solution:

A number is divisible by both 5 and 12

= 60

EXERCISE 3.5

1. Which of the following statements are true?

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

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

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

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

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

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

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

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

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

Solution:

(a) F

 (b) T

 (c) F

(d) T

 (e) F

 (f) F

 (g) T

 (h) T

 (i) F

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

Solution:

(a)

(b) 60 =30*2

30 = 10*3

10 = 5*2

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

Solution:

Let prime number = 5

Prime factors = 5*1

Composite number = 10

Composite factors = 2*5

            One and number itself

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

Solution:

The greatest 4-digit number = 9999

Prime factors of 9999

3*3*11*101 = 9999.

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

Solution:

The smallest 5-digit number = 10000

Prime factors of 10000

2*2*2*2*5*5*5*5 = 10000

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 number of 1729

= 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:

First example:

Let three consecutive number = 1,2,3

1*2*3 = 6

Divisible by 6

6/6 = 1

Second example:

Let three consecutive number = 2,3,4

2*3*4 = 24

Divisible by 6

24/6 = 4

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

Consecutive odd number = 1,3,5,7,9,11,13,________

Frist example:

Sum of two consecutive odd number

1+3 = 4

Divisible by 4

4/4 = 1

Second example:

Sum of two consecutive odd number

3+5 = 8

Divisible by 4

8/4 = 2

Third example:

Sum of two consecutive odd number

5+7 = 12

Divisible by 4

12/4 = 3

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

(a) 24 = 2 × 3 × 4

(b) 56 = 7 × 2 × 2 × 2

(c) 70 = 2 × 5 × 7

(d) 54 = 2 × 3 × 9

Solution:

(b), (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:

Yes

It two co-prime number divide a number then the product of 2 co-prime number will also divide the given number.

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, we cannot say the number must divisible by 4*6 = 24

For example:

12

12÷ 6 = 2

12 ÷ 24 = X

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

Solution:

Let smallest number = X

Four different prime number = 2,3,5,7

So,

2*3*5*7 = 210

Smallest number = 210

EXERCISE 3.6

1. 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

Solution:

(a) 18,48

18 = 2*3*3

48 = 2*2*2*2*3

The common factor of 18,48 are 2 and 3.

Thus, HCF of 18 and 48 are 2*3 = 6.

(b) 30,42

30 = 2*3*5

42 = 2*3*7

The common factor of 30,42 are 2 and 3.

Thus, HCF of 30 and 42 are 2*3 = 6.

(c) 18,60

18 = 2*3*3

60 = 2*2*3*5

The common factor of 18,60 are 2 and 3.

Thus, HCF of 18 and 60 are 2*3 = 6.

(d) 27,63

27 = 3*3*3

63 = 3*3*7

The common factor of 27,63 is 3 (occurring twice).

Thus, HCF of 27and 63 are 3*3 = 9

(e) 36,84

36 = 2*2*3*3

84 = 2*2*3*7

The common factor of 36,84 are 2,2 and 3.

Thus, HCF of 36 and 84 are 2*2*3 = 12.

(f) 34,102

34 = 2*17

102 = 2*3*17

The common factor of 34,102 are 2 and 17.

Thus, HCF of 34 and 102 are 2*17 = 34

(g) 70,105,175

70 = 2*5*7

105 = 3*5*7

175 = 5*5*7

The common factor of 70,105 and 175 are 5 and 7.

Thus, HCF of 70,105 and 175 are 5*7 = 35.

(h) 91,112,49

91 = 7*13

112 = 2*2*2*2*7

49 = 7*7

The common factors of 91,112 and 49 is 7.

Thus, HCF of 91,112 and 49 is 7.

(i) 18,54,81

18 = 2*3*3

54 = 2*3*3*3

81 = 3*3*3*3

The common factor of 18,54 and 81 is 3(occurring twice)

Thus, HCF of 18,54 and 81 are 3*3 = 9.

(j) 12,45,75

12 = 2*2*3

45 = 3*3*5

75 = 3*5*5

The common factor of 12,45 and 75 is 3.

Thus, HCF of 12,45 and 75 is 3.

2. What is the HCF of two consecutive:

 (a) numbers? (b) even numbers? (c) odd numbers?

Solution:

(a) numbers?

HFC of any two consecutive number is always 1.

(b) even number?

HCF of any two even consecutive number is always 2.

(c) odd number?

HCF of any two odd consecutive number is always 1.

3. HCF of co-prime numbers 4 and 15 was found as follows by factorisation: 4 = 2 × 2 and 15 = 3 × 5 since there is no common prime factor, so HCF of 4 and 15 is 0. Is the answer correct? If not, what is the correct HCF?

Solution:

No, 0 is not correct answer because

4 = 1*2*2

15 = 1*3*5

The common factor of 4 and 15 is 1.

Thus, HCF of 4 and 15 is 1.

EXERCISE 3.7

1. Renu purchases two bags of fertilizer of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertilizer exact number of times.

Solution:

Weights of two bags of fertilizer are 75kg and 69kg

75 = 3*5*5

69 = 3*23

The common factor of 75 and 69 is 3.

Thus, HCF of 75 and 69 is 3.

Therefore, maximum value weight 3kg.

It’s the weight of fertilizer exact number of time.

2. 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:

Three boys step off together, the steps measure 63cm, 70cm and 77cm.

Thus, LCM of 63,70 and 77 are

= 7*9*10*11

= 6930 

Minimum distance = 6930

3. 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 length, breath and height of a room are 825cm,675cm and 450cm.

825 = 3*5*5*11

675 = 3*3*3*5*5

450 = 2*3*3*5*5

The common factor of 825, 675 and 450 are 3,5 and 5.

Thus, HCF of 825,675,450 are 3*5*5 = 75.

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

Solution:

Frist we find LCM of 6,8 and 12.

LCM = 2*2*3*2

= 24

The smallest 3-digit number = 120.          {24*1 = 24

                                                                          24*2 = 48

                                                                          24*3 = 72

                                                                          24*4 = 96

                                                                          24*5 =120}

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

Solution:

Frist we find LCM of 8, 10 and 12.

LCM = 2*2*2*3*5

= 120

The greatest 3-digit number = 960.                                    [120*10 = 1200

                                                                                                   120*9 = 1080

                                                                                                   120*8 = 960]

6. 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:

Frist we find LCM of 48, 72 and 108

LCM = 2*2*2*2*3*3*3

= 432 second

= 7 min 12 second at 7a.m.

7. Three tankers contain 403 liters, 434 liters and 465 liters of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

Solution:

Frist we find HCF of 403,434 and 465.

403 = 13*31

434 = 2*7*31

465 = 15*31

The common factor of 403 ,434 and 465.

Thus, HCF of 403, 434 and 465 is 31.

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

Solution:

Frist we find the LCM of 6.15 and 18

LCM = 2*3*3*5

= 90

90 is the least number which when divided by the given numbers will leave remainder 0 in each case. But we need the least number that leaves remainder 5 in each case.

Therefore, the required number is 5 more than 90. The required least number = 90 + 5 = 95.

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

Solution:

Frist we find LCF of 18,24 and 32.

LCM = 2*2*2*2*2*3*3

= 288

The smallest 4-digit number = 1152.              [288*2 = 576

                                                                                 288*3 = 864

                                                                                  288*4 = 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

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

Solution:

In Mathematics, LCM (The Least Common Multiple) of any two is the value that is evenly divisible by the two given numbers.

(a) 9 and 4

9 =3 × 3

4 = 2 × 2

L.C.M of 9 and 4 = 2×2×3×3 = 36

(b) 12 and 5

12 = 2 x 2 x 3

5 = 5 x 1

L.C.M of 12 and 5 = 2×2×3×5 = 60

(c) 6 and 5

6 = 2 × 3

5 = 5× 1

L.C.M of 6 and 5 = 2 × 3 × 5 = 30

(d) 15 and 4

15 = 3× 5

4 = 2 × 2

L.C.M of 15 and 4 = 2 × 2 × 3 × 5 = 60

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.

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

What do you observe in the results obtained?

Solution:

(a) 5,20

LCM = 2*2*5

= 20

(b) 6,18

LCM = 2*3*3

= 18

(c) 12,48

LCM = 2*2*2*2*3

= 48

(d) 9,45

LCM = 3*3*5

= 45