Whole Numbers (Exercise 3.1) RD Sharma Solutions | Mathematics (Maths) Class 6 PDF Download

Q.1. Write down the smallest natural number.

Ans: The smallest natural number is 1.

Q.2. Write down the smallest whole number.

Ans: The smallest whole number is 0 (zero).

Q.3. Write down, if possible, the largest natural number.

Ans: We know that every natural number has a successor. Thus, there is no largest natural number.

Q.4. Write down, if possible, the largest whole number.

Ans: We know that every whole number has a successor. Thus, there is no largest whole number.

Q.5. Are all natural numbers also whole numbers?

Ans: Yes, all natural numbers are whole numbers.

Q.6. Are all whole numbers also natural numbers?

Ans: No, all whole numbers are not natural numbers because 0 is a whole number but not a natural number.

Q.7. Give successor of each of the whole numbers?

(i) 1000909

(ii) 2340900

(iii) 7039999

Ans: 

     Given Number                  Successor

(i) 1,000,909                          1,000,909 + 1 = 1,000,910

(ii) 2,340,900                          2,340,900 + 1 = 2,340,901

(iii) 7,039,999                         7,039,999 + 1 = 7,040,000

Q.8. Write down the predecessor of each of the following whole numbers:

(i) 10000

(ii) 807000

(iii) 7005000

Ans: 

    Given Number                       Predecessor

(i) 10,000                                   10,000 - 1 = 9,999

(ii) 807,000                                 807,000 - 1 = 806,999

(iii) 7,005,000                             7,005,000 - 1 = 7,004,999

Q.9. Represent the following numbers on the number line:

2,0,3,5,7,11,15

Ans: 

Q.10. How many whole numbers are there between 21 and 61?

Ans: The whole numbers between 21 and 61 are 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 and 60.

Thus, there are 39 whole numbers between 21 and 61.

Q.11. Fill in the blanks with the appropriate symbol < or >:

(i) 25...205

(ii) 170...107

(iii) 415...514

(iv) 10001...100001

(v) 2300014...2300041

Ans: We have:

(i) 25 < 205

(ii) 170 > 107

(iii) 415 < 514

(iv) 10001 < 100001

(v) 2300014 < 2300041

Q.12. Arrange the following numbers is descending order:

925, 786, 1100, 141, 325, 886, 0, 270

Ans: Numbers in descending order:

1100, 925, 886, 786, 325, 270, 141, 0

Q.13. Write the largest number of 6 digits and the smallest number of 7 digits. Which one of these two is larger and by how much?

Ans: 

Largest six-digit number = 999,999

Smallest seven-digit number = 1,000,000

Thus, the smallest seven-digit number is larger than the largest six-digit number.

Again,

Difference between these two numbers = 1,000,000 - 999,999 = 1

Hence, the smallest seven-digit number is larger than the largest six-digit number by 1.

Q.14. Write down three consecutive whole numbers just preceding 8510001.

Ans: We have:

First number = 8,510,001 - 1 = 8,510,000

Second number = 8,510,000 - 1 = 8,509,999

Third number = 8,509,999 - 1 = 8,509,998

Hence, the three consecutive whole numbers just preceding 8,510,001 are 8,510,000, 8,509,999 and 85,09,998.

Q.15. Write down the next three consecutive whole numbers starting from 4009998.

Ans: We have:

First number = 4,009,998 + 1 = 4,009,999

Second number = 4,009,999 + 1 = 4,010,000

Third number = 4,010,000 + 1 = 4,010,001

Hence, the next three consecutive whole numbers starting from 4,009,998 will be 4,009,999, 4,010,000 and 4,010,001.

Q.16. Give arguments in support of the statement that there does not exist the largest natural number.

Ans: We know that every natural number has a successor. Therefore, the largest natural number does not exist.

Q.17. Which of the following statements are true and which are false?

(i) Every whole number has its successor.

(ii) Every whole number has its predecessor.

(iii) 0 is the smallest natural number.

(iv) 1 is the smallest whole number.

(v) 0 is less than every natural number.

(vi) Between any two whole numbers there is a whole number.

(vii) Between any two non-consecutive whole numbers there is a whole number.

(viii) The smallest 5-digit number is the successor of the largest 4 digit number

(ix) Of the given two natural numbers, the one having more digits is greater.

(x) The predecessor of a two digit number cannot be a single digit number.

(xi) If a and b are natural numbers and a < b, than there is a natural number c such that a<b<c.

(xii) If a and b are whole numbers and a<b, then a+1< b+1.

(xiii) The whole number 1 has 0 as predecessor.

(xiv) The natural number 1 has no predecessor.

Ans: 

(i) True

The successor of every whole number can be found by adding 1.

(ii) False

Zero (0) is a whole number whose predecessor (-1) is not a whole number.

(iii) False

1 is the smallest natural number.

(iv) False

Zero (0) is the smallest whole number.

(v) True

The smallest natural number is 1, so zero (0) is less than every natural number.

(vi) False

There is no whole number between two consecutive whole numbers.

(vii) True

(viii) True

The smallest five-digit number = 10,000

The largest four-digit number = 9,999

Difference = 10,000 - 9,999 = 1

Because the difference is 1, 10,000 is the successor of 9,999.

(ix) True

(x) False

10 is a two-digit number whose predecessor is 9, which is a one-digit number.

(xi) False

If a and b are consecutive natural numbers, then there cannot be any natural number c in between a and b.

(xii) True

(xiii) True

(xiv) True

The predecessor of natural number 1 is 0, which is not a natural number.

Objective Type Questions

Q.1. The smallest natural number is

(a) 0

(b) 1

(c) -1

(d) None of these

Ans: (b) 1

Q.2. The smallest whole number is

(a) 1

(b) 0

(c) -1

(d) None of these

Ans: (b) 0

Q.3. The predecessor of 1 in natural numbers is

(a) 0

(b) 2

(c) -1

(d) None of these

Ans: (d) None of these

We know that the smallest natural number is 1. Hence, its predecessor does not exist.

Q.4. The predecessor of 1 in whole numbers is

(a) 0

(b) -1

(c) 2

(d) None of these

Ans: (a) 0

Predecessor of 1 = 1 - 1 = 0

Q.5. The predecessor of 1 million is

(a) 9999

(b) 99999

(c) 999999

(d) 1000001

Ans: (c) 9,99,999

We have:

1 million = 10,00,000

Predecessor of 1 million 

= 10,00,000 - 1

= 9,99,999

Q.6. The successor of 1 million is

(a) 10001

(b) 100001

(c) 1000001

(d) 10000001

Ans: (c) 10,00,001

We have:

1 million = 10,00,000

Successor of 1 million 

= 10,00,000 + 1

= 10,00,001

Q.7. The product of the successor and predecessor of 99 is

(a) 9800

(b) 9900

(c) 1099

(d) 9700

Ans: (a) 9800

We have:

Successor of 99 = 99 + 1 = 100

Predecessor of 99 = 99 − 1 = 98

Their product = 100 × 98 = 9800

Q.8. The product of a whole number (other than zero) and its successor is

(a)  an even number

(b) an odd number

(c) divisible by 4

(d) divisible by 3

Ans: (a) an even number

Example:

Whole number = 1

Successor of 1 = 1 + 1 = 2

Their product = 1 × 2 = 2 

Thus, 2 is an even number.

Q.9. The product of the predecessor and successor of an odd natural number is always divisible by

(a) 2

(b) 4

(c) 6

(d) 8

Ans: (d) 8

The predecessor of an odd number is an even number.

The successor of an odd number is also an even number.

These two even numbers are two consecutive even numbers, and the product of two consecutive even numbers is always divisible by 8.

Q.10. The product of the predecessor and successor of an even natural number is

(a) divisible by 2

(b) divisible by 3

(c) divisible by 4

(d) an odd number

Ans: (d) an odd number

Example:

Even natural number = 2

Predecessor of  2 = 2 − 1 = 1

Successor of 2 = 2 + 1 = 3

Their product = 1 × 3 = 3

Thus, the product is an odd number.

Q.11. The successor of the smallest prime number is

(a) 1

(b) 2

(c) 3

(d) 4

Ans: The smallest prime number is 2

So, Successor of 2 = 2 + 1 = 3

Hence, the correct answer is option (c).

Q.12. If x and y are co-primes, then their LCM is

(a) 1

(b) x/y

(c) xy

(d) None of these

Ans: A set of numbers which do not have any other common factor other than 1 are called co-prime.

The LCM of two co-prime numbers is equal to their product.

Hence, the correct answer is option (c).

Q.13. The HCF of two co-primes is

(a) the smaller number

(b) the larger number

(c) product of the numbers

(d) 1

Ans: A set of numbers which do not have any other common factor other than 1 are called co-prime.

The HCF of two co-prime numbers is 1.

Hence, the correct answer is option (d).

Q.14. The smallest number which is neither prime nor composite is

(a) 0

(b) 1

(c) 2

(d) 3

Ans: The smallest number which is neither prime nor composite is 1

Hence, the correct answer is option (b).

Q.15. The product of any natural number and the smallest prime is

(a) an even number

(b) an odd number

(c) a prime number

(d) None of these

Ans: The smallest prime number is 2.

Thus, when we multiply any natural number we will always get an even number.

Hence, the correct answer is option (a).

Q.16. Every counting number has an infinite number of

(a) factors

(b) multiples

(c) prime factors

(d) None of these

Ans: Multiples are what we get after multiplying the number by any number.

Thus, every counting number has an infinite number of multiples

Hence, the correct answer is option (b).

Q.17. The product of two numbers is 1530 and their HCF is 15. The LCM of these numbers is

(a) 102

(b) 120

(c) 84

(d) 112

Ans: 

Product of two numbers = HCF of two numbers × LCM of two numbers

⇒1530=15×LCM of two numbers

⇒LCM of two numbers== 102

Hence, the correct answer is option (a).

Q.18. The least number divisible by each of the numbers 15, 20, 24 and 32 is

(a) 960

(b) 480

(c) 360

(d) 640

Ans: LCM of 15, 20, 24 and 32 is given by

15 = 3 × 5 = 3¹ × 5¹

20 = 2 × 2  × 5 = 2²  × 5¹

24 = 2 × 2  × 2 × 3 = 2³  × 3¹

32 = 2 × 2  × 2 × 2  × 2 = 2⁵

LCM = 2⁵ × 3¹  × 5¹ = 480

Hence, the correct answer is option (b).

Q.19. The greatest number which divides 134 and 167 leaving 2 as remainder in each case is

(a) 14

(b) 19

(c) 33

(d) 17

Ans: First we subtract the required remainder from 134 and 167.

Thus, we will get 132 and 165.

132 = 2 × 2 × 3 × 11 = 2² × 3 × 11

165 = 3 × 5 × 11 = 3¹ × 5 × 11

HCF = 3 × 11 = 33

Thus, the greatest number which divides 134 and 167 leaving 2 as remainder in each case is 33

Hence, the correct answer is option (c).

Q.20. Which of the following numbers is a prime number?

(a) 91

(b) 81

(c) 87

(d) 97

Ans: Since, factors of

91 = 1 × 7 × 13 

81 = 1 × 3 × 3 × 3 × 3

87 = 1 × 3 × 29

97 = 1 × 97

Thus, 81, 87 and 91 all are not prime numbers.

Hence, the correct answer is option (d).

Q.21. If two numbers are equal, then

(a) their LCM is equal to their HCF

(b) their LCM is less than their HCF

(c) their LCM is equal to two times their HCF

(d) None of these

Ans: If two numbers are equal, then their LCM is equal to their HCF

Hence, the correct answer is option (a).

Q.22. a and b are two co-primes. Which of the following is/are true?

(a) LCM (a, b) = a × b

(b) HCF (a, b) = 1

(c) Both (a) and (b)

(d) Neither (a) nor (b)

Ans: A set of numbers which do not have any other common factor other than 1 are called co-prime.

The LCM of two co-prime numbers is equal to their product.

The HCF of two co-prime numbers is 1.

Hence, the correct answer is option (c).