Natural Numbers & Whole Numbers

(i) The smallest natural number is 1.

(ii) The smallest whole number is 0.

(iii) The largest natural number is not possible (natural numbers are infinite).

(iv) The largest whole number is not possible (whole numbers are infinite).

(v) All natural numbers are whole numbers.

(vi) All whole numbers are not natural numbers.

(vii) Successor of 4099 = 4099 + 1 = 4100.

(viii) Predecessor of 4330 = 4330 - 1 = 4329.

(i) True. The sum of two whole numbers is always a whole number, so whole numbers are closed for addition.

(ii) False. By the closure property of addition, a + b is always a whole number.

(iii) True. Addition of whole numbers is commutative, so a + b = b + a.

(iv) True. By the commutative property, 0 + 18 = 18 + 0 = 18.

(v) True. Addition of whole numbers is associative.

(vi) True. By the associative property, (10 + 12) + 16 = 10 + (12 + 16) = 38.

(i) By the Commutative Property of Addition: a + b = b + a.

54 + 234 = 234 + 54.

(ii) By the Commutative Property of Addition: a + b = b + a.

332 + 497 = 497 + 332.

(iii) By the Additive Identity property, adding 0 to a whole number gives the number itself.

286 + 0 = 286.

(iv) By the Multiplicative Identity property, multiplying a whole number by 1 gives the number itself.

286 × 1 = 286.

(v) By the Associative Property of Addition: a + (b + c) = (a + b) + c.

a + (b + c) = (a + b) + c.

(i) 3 + (5 + 4) = (3 + 5) + 4

L.H.S. = 3 + (5 + 4) = 3 + 9 = 12.

R.H.S. = (3 + 5) + 4 = 8 + 4 = 12.

Since L.H.S. = R.H.S., the result is verified.

(ii) 8 × (8 + 0) = 8 × 8 + 8 × 0

L.H.S. = 8 × (8 + 0) = 8 × 8 = 64.

R.H.S. = 8 × 8 + 8 × 0 = 64 + 0 = 64.

Since L.H.S. = R.H.S., the result is verified.

(iii) (7 + 6) × 10 = 7 × 10 + 6 × 10

L.H.S. = (7 + 6) × 10 = 13 × 10 = 130.

R.H.S. = 7 × 10 + 6 × 10 = 70 + 60 = 130.

Since L.H.S. = R.H.S., the result is verified.

(iv) (15 - 12) × 18 = 15 × 18 - 12 × 18

L.H.S. = (15 - 12) × 18 = 3 × 18 = 54.

R.H.S. = 15 × 18 - 12 × 18 = 270 - 216 = 54.

Since L.H.S. = R.H.S., the result is verified.

(v) 16 + 0 = 16

L.H.S. = 16 + 0 = 16 = R.H.S.

By the Additive Identity property, the result is verified.

(vi) 23 + (-23) = 0

L.H.S. = 23 + (-23) = 23 - 23 = 0 = R.H.S.

The result is verified.

(i) False. The sum of two odd numbers is an even number. For example, 3 + 5 = 8.

(ii) True. The sum of two odd numbers is an even number. For example, 7 + 9 = 16.

(iii) True. The sum of two even numbers is an even number. For example, 4 + 6 = 10.

(iv) False. The sum of two even numbers is an even number, not an odd number.

(v) True. The sum of an even number and an odd number is an odd number. For example, 4 + 3 = 7.

(vi) False. 0 is a whole number but it is not a natural number, so every whole number is not a natural number.

(vii) True. Every natural number is a whole number.

(viii) True. Adding 0 to any whole number gives the number itself (additive identity).

(ix) True. Multiplying any whole number by 1 gives the number itself (multiplicative identity).

(x) False. Commutativity and associativity are not properties of every operation (for example, they do not hold for subtraction and division). The statement is incomplete as it does not mention the operation.

(xi) True. For the operation of addition, both commutativity and associativity hold for natural numbers as well as whole numbers.

(xii) False. If x is a whole number, then -x is a negative number, and negative numbers are not whole numbers.

(i) Since a is greater than b, the difference a - b is a positive number.

So, a - b is a whole number, and yes, this result is always true (whenever a is greater than b).

(ii) Since a is greater than b, the difference b - a is a negative number, and negative numbers are not whole numbers.

So, b - a is not a whole number, and yes, this result is always true (whenever a is greater than b, b - a is always negative).

For any whole number x, we have x - 0 = x, but 0 - x ≠ x.

An identity element must give the same number when applied from both sides, but for subtraction 0 works only from the right side and not from the left.

Hence, the identity number for subtraction of whole numbers is not possible (it does not exist).

(i) 12 × (9 - 6) = 12 × 3 = 36.

(ii) 12 × 9 - 12 × 6 = 108 - 72 = 36.

(iii) Both results are equal to 36, so yes, 12 × (9 - 6) = 12 × 9 - 12 × 6.

(iv) Yes. By the Distributive Law of Multiplication over Subtraction, this type of result is always true.

(i) (16 - 8) × 24 = 8 × 24 = 192.

(ii) 16 × 24 - 8 × 24 = 384 - 192 = 192.

(iii) Both results are equal to 192, so yes, (16 - 8) × 24 = 16 × 24 - 8 × 24.

(iv) Yes. By the Distributive Law of Multiplication over Subtraction, this type of result is always true.

The largest number of four digits = 9999.

The smallest number of six digits = 100000.

Difference = 100000 - 9999 = 90001.

Hence, the required difference is 90001.

The smallest number of eight digits = 10000000.

The largest number of five digits = 99999.

Difference = 10000000 - 99999 = 9900001.

Hence, the required difference is 9900001.

(i) Any number multiplied by 0 gives 0.

42 × 0 = 0.

(ii) Any number multiplied by 1 gives the number itself.

592 × 1 = 592.

(iii) By the Commutative Property of Multiplication: a × b = b × a.

328 × 573 = 573 × 328.

(iv) By the Commutative Property of Multiplication: a × b = b × a.

229 × 578 = 578 × 229.

(v) Since 6 + 7 + 2 = 15, we have:

32 × 15 = 32 × 6 + 32 × 7 + 32 × 2.

(vi) Since 23 = 20 + 3, we have:

23 × 56 = 20 × 56 + 3 × 56.

(vii) Using the Distributive Law of Multiplication over Addition:

83 × 54 + 83 × 16 = 83 × (54 + 16) = 83 × 70 = 5810.

(viii) Using the Distributive Law of Multiplication over Subtraction:

98 × 273 - 75 × 273 = (98 - 75) × 273 = 23 × 273.

(i) 984 × 102

⇒ 984 × (100 + 2)

Using the Distributive Law of Multiplication over Addition:

⇒ 984 × 100 + 984 × 2

⇒ 98400 + 1968

⇒ 100368.

Hence, 984 × 102 = 100368.

(ii) 385 × 1004

⇒ 385 × (1000 + 4)

Using the Distributive Law of Multiplication over Addition:

⇒ 385 × 1000 + 385 × 4

⇒ 385000 + 1540

⇒ 386540.

Hence, 385 × 1004 = 386540.

(iii) 446 × 10002

⇒ 446 × (10000 + 2)

Using the Distributive Law of Multiplication over Addition:

⇒ 446 × 10000 + 446 × 2

⇒ 4460000 + 892

⇒ 4460892.

Hence, 446 × 10002 = 4460892.

(i) 548 × 98

⇒ 548 × (100 - 2)

Using the Distributive Law of Multiplication over Subtraction:

⇒ 548 × 100 - 548 × 2

⇒ 54800 - 1096

⇒ 53704.

Hence, 548 × 98 = 53704.

(ii) 924 × 988

⇒ 924 × (1000 - 12)

Using the Distributive Law of Multiplication over Subtraction:

⇒ 924 × 1000 - 924 × 12

⇒ 924000 - 11088

⇒ 912912.

Hence, 924 × 988 = 912912.

(i) 679 × 8 + 679 × 2

Using the Distributive Law of Multiplication over Addition: a × b + a × c = a × (b + c).

⇒ 679 × (8 + 2)

⇒ 679 × 10

⇒ 6790.

Hence, 679 × 8 + 679 × 2 = 6790.

(ii) 284 × 12 - 284 × 2

Using the Distributive Law of Multiplication over Subtraction: a × b - a × c = a × (b - c).

⇒ 284 × (12 - 2)

⇒ 284 × 10

⇒ 2840.

Hence, 284 × 12 - 284 × 2 = 2840.

(iii) 55873 × 94 + 55873 × 6

Using the Distributive Law of Multiplication over Addition:

⇒ 55873 × (94 + 6)

⇒ 55873 × 100

⇒ 5587300.

Hence, 55873 × 94 + 55873 × 6 = 5587300.

(iv) 7984 × 15 - 7984 × 5

Using the Distributive Law of Multiplication over Subtraction:

⇒ 7984 × (15 - 5)

⇒ 7984 × 10

⇒ 79840.

Hence, 7984 × 15 - 7984 × 5 = 79840.

(v) 8324 × 1945 - 8324 × 945

Using the Distributive Law of Multiplication over Subtraction:

⇒ 8324 × (1945 - 945)

⇒ 8324 × 1000

⇒ 8324000.

Hence, 8324 × 1945 - 8324 × 945 = 8324000.

(vi) 3333 × 987 + 13 × 3333

⇒ 3333 × 987 + 3333 × 13

Using the Distributive Law of Multiplication over Addition:

⇒ 3333 × (987 + 13)

⇒ 3333 × 1000

⇒ 3333000.

Hence, 3333 × 987 + 13 × 3333 = 3333000.

(i) The greatest number of three digits = 999.

The smallest number of five digits = 10000.

Product = 999 × 10000 = 9990000.

Hence, the required product is 9990000.

(ii) The greatest number of four digits = 9999.

The greatest number of three digits = 999.

Product = 9999 × 999

⇒ 9999 × (1000 - 1)

⇒ 9999 × 1000 - 9999 × 1

⇒ 9999000 - 9999

⇒ 9989001.

Hence, the required product is 9989001.

(i) Here, 437 + 3 = 440 and 400 - 3 = 397.

So, (437 + 3) × (400 - 3) = 440 × 397 = 397 × 440 [by commutative property]

⇒ 397 × 440 = 174680.

(ii) Since 66 = 11 × 6, 44 = 11 × 4 and 22 = 11 × 2, using the distributive property:

⇒ 66 + 44 + 22 = 11 × 6 + 11 × 4 + 11 × 2

⇒ 11 × (6 + 4 + 2) = 11 × 12 = 132.

(i) Division of whole numbers is not closed because the result of dividing one whole number by another is not always a whole number.

Examples: 5 ÷ 8 and 12 ÷ 24 are not whole numbers, etc.

(ii) Any whole number divided by 1, always gives the number itself.

Examples: 5 ÷ 1 = 5, and 16 ÷ 1 = 16, etc.

(iii) Every non-zero whole number divided by itself gives 1.

Examples: 8 ÷ 8 = 1, and 12 ÷ 12 = 1, etc.

(iv) Zero divided by any non-zero whole number gives zero only.

Examples: 0 ÷ 6 = 0, and 0 ÷ a = 0, if a ≠ 0, etc.

(v) A whole number divided by 0 is not defined.

Examples: 7 ÷ 0 is not defined, and 16 ÷ 0 is not defined, etc.

For every non-zero whole number, dividing it by itself gives 1, i.e. x ÷ x = 1.

So, x ÷ x = x means x = 1.

Hence, the value of x is 1.

(i) Any whole number divided by 1 gives the number itself.

987 ÷ 1 = 987.

(ii) Zero divided by any non-zero whole number gives 0.

0 ÷ 987 = 0.

(iii) Any non-zero whole number divided by itself gives 1.

⇒ 888 ÷ 888 = 1

⇒ 336 - (888 ÷ 888) = 336 - 1 = 335.

(iv) Any non-zero whole number divided by itself gives 1.

⇒ 23 ÷ 23 = 1 and 437 ÷ 437 = 1

⇒ (23 ÷ 23) - (437 ÷ 437) = 1 - 1 = 0.

(i) L.H.S. = 12 ÷ (6 × 2) = 12 ÷ 12 = 1.

R.H.S. = (12 ÷ 6) × (12 ÷ 2) = 2 × 6 = 12.

Since L.H.S. ≠ R.H.S., this statement is false.

(ii) Division is not distributive over subtraction in this form, so a ÷ (b - c) ≠ $\frac{a}{b} - \frac{a}{c}$.

This statement is false.

(iii) (a - b) ÷ c = $\frac{a - b}{c} = \frac{a}{c} - \frac{b}{c}$.

This statement is true.

(iv) L.H.S. = (15 - 13) ÷ 8 = 2 ÷ 8 = $\frac{2}{8} = \frac{1}{4}$.

R.H.S. = (15 ÷ 8) - (13 ÷ 8) = $\frac{15}{8} - \frac{13}{8} = \frac{2}{8} = \frac{1}{4}$.

Since L.H.S. = R.H.S., this statement is true.

(v) L.H.S. = 8 ÷ (15 - 13) = 8 ÷ 2 = 4.

R.H.S. = $\frac{8}{15} - \frac{8}{13}$, which is a negative number.

Since L.H.S. ≠ R.H.S., this statement is false.

Hence, statements (iii) and (iv) are true.

(i) The next three steps are:

1234 × 9 + 4 = 11110

12345 × 9 + 5 = 111110

123456 × 9 + 6 = 1111110

(ii) The next three steps are:

9876 × 9 + 4 = 88888

98765 × 9 + 3 = 888888

987654 × 9 + 2 = 8888888

(iii) The next three steps are:

1234 × 8 + 4 = 9876

12345 × 8 + 5 = 98765

123456 × 8 + 6 = 987654

(iv) The next three steps are:

444 ÷ 12 = 37

555 ÷ 15 = 37

666 ÷ 18 = 37

In a magic square, the sum of the numbers in each row, each column and each diagonal is the same. This common sum is called the magic sum, and for a 3 × 3 magic square it equals 3 times the centre number.

(i) The centre number is 5, so the magic sum = 3 × 5 = 15.

Row 1: 6 + 7 + ............... = 15 ⇒ missing number = 2.

Row 2: ............... + 5 + 9 = 15 ⇒ missing number = 1.

Row 3: 8 + ............... + 4 = 15 ⇒ missing number = 3.

The missing numbers (row-wise) are 2, 1 and 3. The completed magic square is:

6	7	2
1	5	9
8	3	4

(ii) The centre number is 7, so the magic sum = 3 × 7 = 21.

Diagonal: 8 + 7 + ............... = 21

⇒ Bottom-left number = 6.

Row 1: 4 + ............... + 8 = 21

⇒ Missing number = 9.

Diagonal 2: 8 + 7 + ............... = 21

⇒ Missing number = 6.

Column 1: 4 + ............... + 6 = 21

⇒ Middle-left number = 11.

Row 2: 11 + 7 + ............... = 21

⇒ Middle-right number = 3.

Row 3: 6 + ............... + 10 = 21

⇒ Missing number = 5.

The missing numbers (row-wise) are 9, 11, 3, 6 and 5. The completed magic square is:

4	9	8
11	7	3
6	5	10

(iii) The centre number is 10, so the magic sum = 3 × 10 = 30.

Row 1: 16 + 2 + ............... = 30

⇒ Missing number = 12.

Diagonal: 12 + 10 + ............... = 30

⇒ Bottom-left number = 8.

Column 1: 16 + ............... + 8 = 30

⇒ Middle-left number = 6.

Row 2: 6 + 10 + ............... = 30

⇒ Middle-right number = 14.

Row 3: 8 + ............... + 4 = 30

⇒ Missing number = 18.

The missing numbers (row-wise) are 12, 6, 14, 8 and 18. The completed magic square is:

16	2	12
6	10	14
8	18	4

From the pattern, the number of matchsticks for the first few figures is:

Figure number (n)	Number of matchsticks (S)
1	7
2	10
3	13
4	16

(i) Each new figure is formed by adding one square, and each added square shares one edge with the previous square.

So, each new figure requires 3 more matchsticks.

Therefore, for the nth figure,

S = 7 + 3(n - 1)

S = 7 + 3n - 3

S = 3n + 4

Hence, S = 3n + 4.

(ii) Calculating,

(1) For the 15th figure, n = 15:

S = 3 × 15 + 4 = 45 + 4 = 49 matchsticks.

(2) For the 40th figure, n = 40:

S = 3 × 40 + 4 = 120 + 4 = 124 matchsticks.

(iii) The number of matchsticks (S) is equal to 4 more than three times the figure number (n).

(i) The next two figures in the pattern are:

(ii) The table describing the figures is:

Number of figures (n)	1	2	3	4	5
Number of matchsticks (L)	2	4	6	8	10

(iii) Each time the number of figures (n) increases by 1, the number of matchsticks (L) increases by 2.

For n = 1, 2n = 2 and L = 2.

Hence, L = 2n.

(iv) (1) For the 12th figure, n = 12:

L = 2 × 12 = 24 matchsticks.

(2) For the 20th figure, n = 20:

L = 2 × 20 = 40 matchsticks.

The predecessor of a whole number is obtained by subtracting 1 from it.

The predecessor of 0 would be 0 - 1 = -1, which is not a whole number.

So, 0 is the only whole number that does not have a predecessor.

Hence, option 2 is the correct option.

Ten thousand = 10000.

Predecessor of 10000 = 10000 - 1 = 9999.

Hence, option 2 is the correct option.

Using the Distributive Law of Multiplication over Addition:

43 × 27 + 43 × 73 = 43 × (27 + 73)

= 43 × 100 = 4300.

Hence, option 1 is the correct option.

484 × 101 = 484 × (100 + 1)

Using the Distributive Law of Multiplication over Addition:

= 484 × 100 + 484 × 1

= 48400 + 484 = 48884.

Hence, option 3 is the correct option.

Using the Distributive Law of Multiplication over Subtraction:

57 × 13 - 57 × 3 = 57 × (13 - 3) = 57 × 10.

Hence, option 2 is the correct option.

Every non-zero whole number divided by itself gives 1.

(17 ÷ 17) - (97 ÷ 97) = 1 - 1 = 0.

Hence, option 3 is the correct option.

The whole numbers between 3 and 7 are 4, 5 and 6.

The correct number line is the one in which only the points 4, 5 and 6 are marked between 3 and 7.

Hence, option 2 is the correct option.

The greatest number of two digits = 99.

The smallest number of two digits = 10.

Product = 99 × 10 = 990.

Hence, option 2 is the correct option.

For every non-zero whole number, a ÷ a = 1.

So, a ÷ a = a means a = 1.

Hence, option 3 is the correct option.

L.H.S. = 8 - (a - 7) = 8 - a + 7 = 15 - a.

R.H.S. = 8 + (4 - b) = 8 + 4 - b = 12 - b.

So, 15 - a = 12 - b, which gives b - a = -3.

Substituting values from option 2:

⇒ a = -4 and b = -7 gives b - a = -7 - (-4) = -3.

Hence, option 2 is the correct option.

Statement 1: x - 0 = x shows that 0 is a right identity for subtraction. But 0 is an identity element only if it works from both the right and the left, i.e. 0 - x = x must also hold. Since 0 - x ≠ x, the number 0 is not the identity element for subtraction.

∴ Statement 1 is false.

Statement 2: 0 - x = x is not true for whole numbers (for example, 0 - 5 = -5 ≠ 5), so 0 is not the identity element for subtraction.

∴ Statement 2 is false.

Hence, option 2 is the correct option.

Statement 1: Subtracting one whole number from another does not always give a natural number. For example, 5 - 5 = 0 (which is not a natural number) and 5 - 8 = -3 (which is not even a whole number).

∴ Statement 1 is false.

Statement 2: 15 - 25 = -10, which is a negative number and hence not a natural number. So the statement is correct.

∴ Statement 2 is true.

Hence, option 4 is the correct option.

Assertion (A): Addition of natural numbers is associative, i.e. for natural numbers a, b and c, a + (b + c) = (a + b) + c. So, the assertion is true.

Reason (R): The relation a + (b + c) = (a + b) + c is exactly the statement of the associative property of addition of natural numbers. So, the reason is true and it correctly explains the assertion.

Both A and R are true.

Hence, option 3 is the correct option.

Assertion (A): Using the distributive law, 656 × 42 - 656 × 17 = 656 × (42 - 17) = 656 × 25 = 16400, not 9850. So, the assertion is false.

Reason (R): For any three numbers x, y and z, x × (y - z) = x × y - x × z, which is the distributive property of multiplication over subtraction. So, the reason is true.

A is false and R is true.

Hence, option 2 is the correct option.