Playing With Numbers

1. Factors and Multiples

• Factors: A factor of a number is an exact divisor of that number. Every number (except 1) has at least two factors: 1 and the number itself.
• Multiples: A multiple of a number is obtained by multiplying it by a natural number (1, 2, 3, ……..).
  • Example: Factors of 6 are 1, 2, 3, 6. Multiples of 6 are 6, 12, 18, 24…

3. Tests of Divisibility

These rules allow you to check if a number is divisible by another without full division:

Divisibility by 11: Difference between the sum of digits at odd places and even places is either 0 or a multiple of 11.2. Classification of Numbers

Divisibility by 2: Last digit is 0, 2, 4, 6, or 8.

Divisibility by 3: Sum of the digits is divisible by 3.

Divisibility by 4: Last two digits are divisible by 4.

Divisibility by 5: Last digit is 0 or 5.

Divisibility by 6: Divisible by both 2 and 3.

Divisibility by 9: Sum of the digits is divisible by 9.

Divisibility by 10: Last digit is 0.

• Even and Odd: Even numbers are divisible by 2; odd numbers are not.
• Prime Numbers: Numbers with exactly two factors (1 and itself).
  • Note: 2 is the only even prime number.
• Composite Numbers: Numbers with more than two factors.
• Perfect Numbers: A number where the sum of all its factors (excluding the number itself) is equal to the number. (e.g., 6: 1 + 2 + 3 = 6).
• Twin Primes: Two prime numbers with a difference of 2 (e.g., 3 and 5).
• Co-primes: Two numbers that have only 1 as a common factor (e.g., 8 and 15).

4. Prime Factorization

Expressing a composite number as a product of its prime factors.

• Methods: Factor Tree Method and Division Method.

5. H.C.F. and L.C.M.

• H.C.F. (Highest Common Factor): The largest number that divides two or more numbers exactly. Found using listing factors, prime factorization, or continued division.
• L.C.M. (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers. Found using listing multiples or common division.
• The Formula: H.C.F.* L.C.M. = Product of the two numbers

6. BODMAS Rule

To simplify expressions involving multiple operations, follow this order:

1. Brackets ( (), [], {} )
2. Of (Multiplication)
3. Division
4. Multiplication
5. Addition
6. Subtraction

EXERCISE 9 (A)

(Using BODMAS)
Question 1.
19 – (1 + 5) – 3

Solution :

Expression: 19 – (1 + 5) – 3

Step 1: Solve the Bracket (B)

Inside the bracket, we have 1 + 5.

1 + 5 = 6

So, the expression becomes:

19 – 6 – 3

Step 2: Perform Subtraction (S)

Now, perform the subtractions from left to right.

19 – 6 = 13

The expression becomes:

13 – 3

Step 3: Final Subtraction

13 – 3 = 10

Final Answer: 10

Question 2.
30 x 6 + (5 – 2)

Solution :

Expression: $30 \times 6 + (5 – 2)$

Step 1: Solve the Bracket (B)

Inside the bracket, we have $5 – 2$.

$5 – 2 = 3$

So, the expression becomes:

$30 \times 6 + 3$

Step 2: Perform Multiplication (M)

According to BODMAS, multiplication comes before addition.

30 * 6 = 180

The expression becomes:

180 + 3

Step 3: Perform Addition (A)

$180 + 3 = 183$

Final Answer: 183

Question 3.
28 – (3 x 8) + 6

Solution :

Expression: 28 – (3 * 8) + 6

Step 1: Solve the Bracket (B)

Inside the bracket, we have 3 * 8.

3 * 8 = 24

So, the expression becomes:

28 – 24 + 6

Step 2: Perform Addition and Subtraction (A & S)

In the BODMAS rule, Addition and Subtraction have the same priority. You should work from left to right.

• First, 28 – 24 = 4
• Then, 4 + 6 = 10

Final Answer: 10

Question 4.
9 – [(4 – 3) + 2 x 5]

Solution :

Expression: 9 – [(4 – 3) + 2 * 5]

Step 1: Solve the innermost bracket (Parentheses)

Inside (4 – 3), the value is 1.

The expression now looks like this:

9 – [1 + 2 * 5]

Step 2: Solve inside the square bracket [ ]

Inside the square bracket, we have addition and multiplication. According to BODMAS, Multiplication (M) comes before Addition (A).

• Multiply: 2 * 5 = 10
• The expression becomes: 9 – [1 + 10]

Step 3: Complete the square bracket

Add the numbers inside:

1 + 10 = 11

The expression becomes:

9 – 11

Step 4: Final Subtraction

9 – 11 = -2

Final Answer: -2

Question 5.
[18 – (15 – 5) + 6]

Solution :

Expression: [18 – (15 – 5) + 6]

Step 1: Solve the innermost bracket (Parentheses)

First, we look at the numbers inside the round brackets $( 15 – 5 ).

• 15 – 5 = 10

Now the expression becomes:

[18 – 10 + 6]

Step 2: Solve inside the square bracket [ ]

Inside the square bracket, we have subtraction and addition. According to the rules of addition and subtraction, we solve from left to right.

• First, 18 – 10 = 8
• Then, 8 + 6 = 14

Final Answer: 14

Question 6.
[(4 x 2) – (4 + 2)] + 8

Solution :

Expression: [(4 * 2) – (4 + 2)] + 8

Step 1: Solve the innermost brackets (Parentheses)

There are two sets of round brackets inside the square brackets. We solve both:

• (4 * 2) = 8
• (4 + 2) = 6

Now, the expression looks like this:

[8 – 6] + 8

Step 2: Solve inside the square bracket [ ]

Now we perform the subtraction inside the square bracket:

• 8 – 6 = 2

The expression becomes:

2 + 8

Step 3: Final Addition

• 2 + 8 = 10

Final Answer: 10

Question 7.
48 + 96 – 24 – 6 x 18

Solution :

Expression: 48 + 96 – 24 – 6 * 18

Step 1: Perform Multiplication (M)

First, we look for multiplication. We have 6 * 18.

• 6 * 18 = 108

Now, the expression looks like this:

48 + 96 – 24 – 108

Step 2: Perform Addition and Subtraction (A & S)

Now we are left with addition and subtraction. We solve these from left to right.

• First addition: 48 + 96 = 144 The expression becomes: 144 – 24 – 108
• Next subtraction: 144 – 24 = 120 The expression becomes: 120 – 108
• Final subtraction: 120 – 108 = 12

Final Answer: 12

Question 8.
22 – [3 – {8 – (4 + 6)}]

Solution :

Expression: 22 – [3 – {8 – (4 + 6)}]

Step 1: Solve the innermost Round Bracket ( )

Inside the round brackets, we have 4 + 6.

• 4 + 6 = 10

The expression now looks like this:

22 – [3 – {8 – 10}]

Step 2: Solve the Curly Braces { }

Inside the curly braces, we have 8 – 10.

• 8 – 10 = -2

The expression now looks like this:

22 – [3 – (-2)]

Step 3: Solve the Square Bracket [ ]

Inside the square bracket, we have 3 – (-2). Remember that subtracting a negative number is the same as adding a positive number (- and – make +).

• 3 + 2 = 5

The expression becomes:

22 – 5

Step 4: Final Subtraction

• 22 – 5 = 17

Final Answer: 17

Question 9.

Solution :

Step 1: Solve the Vinculum (Bar)

The bar is over 28 – 26. We solve this first.

• 28 – 26 = 2The expression becomes: 34 – [29 – {30 + 66 / (24 – 2)}]

Step 2: Solve the innermost Round Bracket ( )

Inside the round bracket, we have 24 – 2.

• 24 – 2 = 22 The expression becomes: 34 – [29 – {30 + 66 / 22}]

Step 3: Solve the Curly Braces { }

Inside the curly braces, we have addition and division. According to BODMAS, Division (D) comes before Addition (A).

• Divide: 66 / 22 = 3
• Add: 30 + 3 = 33 The expression becomes: 34 – [29 – 33]

Step 4: Solve the Square Bracket [ ]

Inside the square bracket, we have 29 – 33.

• 29 – 33 = -4The expression becomes: 34 – (-4)

Step 5: Final Subtraction

Remember that subtracting a negative number is the same as adding (34 – (-4) = 34 + 4).

• 34 + 4 = 38

Final Answer: 38

Question 10.
60 – {16 + (4 x 6 – 8)}

Solution :

Step 1: Solve the innermost Round Bracket ( )

Inside the round bracket, we have both multiplication and subtraction. According to BODMAS, Multiplication (M) comes before Subtraction (S).

• Multiply: 4 * 6 = 24
• Subtract: 24 – 8 = 16 The expression now looks like this: 60 – {16 + 16}

Step 2: Solve the Curly Braces { }

Now we add the numbers inside the curly braces.

• 16 + 16 = 32 The expression becomes: 60 – 32

Step 3: Final Subtraction

• 60 – 32 = 28

Final Answer: 28

Question 11.

Solution :

Expression: 25 – [12 – {5 + 18 / (4 – {5 – 3})}]

Step 1: Solve the Vinculum (overline {Bar})

The bar is over 5 – 3.

• 5 – 3 = 2 The expression becomes: 25 – [12 – {5 + 18 / (4 – 2)}]

Step 2: Solve the Round Bracket ( )

Inside the round bracket, we have 4 – 2.

• 4 – 2 = 2 The expression becomes: 25 – [12 – {5 + 18 / 2}]

Step 3: Solve the Curly Braces { }

Inside the curly braces, we have addition and division. According to BODMAS, Division (D) comes before Addition (A).

• Divide: 18 / 2 = 9
• Add: 5 + 9 = 14 The expression becomes: 25 – [12 – 14]

Step 4: Solve the Square Bracket [ ]

Inside the square bracket, we have 12 – 14.

• 12 – 14 = -2 The expression becomes: 25 – (-2)

Step 5: Final Subtraction

When you see two minus signs together (-(-)), they become a plus sign.

• 25 + 2 = 27

Final Answer: 27

Question 12.
15 – [16 – {12 + 21 ÷ (9 – 2)}]

Solution :

Expression: 15 – [16 – {12 + 21 / (9 – 2)}]

Step 1: Solve the innermost Round Bracket ( )

Inside the round bracket, we have 9 – 2.

• 9 – 2 = 7 The expression now looks like this:15 – [16 – {12 + 21 / 7}]

Step 2: Solve the Curly Braces { }

Inside the curly braces, we have addition and division. According to BODMAS, Division (D) must be done before Addition (A).

• Divide: 21 / 7 = 3
• Add: 12 + 3 = 15 The expression now looks like this : 15 – [16 – 15]

Step 3: Solve the Square Bracket [ ]

Now we perform the subtraction inside the square bracket.

• 16 – 15 = 1 The expression becomes: 15 – 1

Step 4: Final Subtraction

• 15 – 1 = 14

Final Answer: 14

EXERCISE 9 (B)

Question 1.
Fill in the blanks :
(i) On dividing 9 by 7, quotient = …………. and remainder = ……….
(ii) On dividing 18 by 6, quotient = …………. and remainder = ………….
(iii) Factor of a number is ………….. of …………..
(iv) Every number is a factor of …………….
(v) Every number is a multiple of …………..
(vi) …………. is factor of every number.
(vii) For every number, its factors are ………… and its multiples are …………..
(viii) x is a factor of y, then y is a ………… of x.

Solution :

(i) On dividing 9 by 7, quotient = 1 and remainder = 3
(ii) On dividing 18 by 6, quotient = 3 and remainder = 0
(iii) Factor of a number is an exact division of the number
(iv) Every number is a factor of itself
(v) Every number is a multiple of itself
(vi) One is factor of every number.
(vii) For every number, its factors are finite and its multiples are infinite
(viii) x is a factor of y, then y is a multiple of x.

Question 2.
Write all the factors of :
(i) 16
(ii) 21
(iii) 39
(iv) 48
(v) 64
(vi) 98

Solution :

i) 16

• 1 * 16 = 16
• 2 * 8 = 16
• 4 * 4 = 16
• Factors of 16 are: 1, 2, 4, 8, 16.

(ii) 21

• 1 * 21 = 21
• 3 * 7 = 21
• Factors of 21 are: 1, 3, 7, 21.

(iii) 39

• 1 * 39 = 39
• 3 * 13 = 39
• Factors of 39 are: 1, 3, 13, 39.

(iv) 48

• 1 * 48 = 48
• 2 * 24 = 48
• 3 * 16 = 48
• 4 * 12 = 48
• 6 * 8 = 48
• Factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

(v) 64

• 1 * 64 = 64
• 2 * 32 = 64
• 4 * 16 = 64
• 8 * 8 = 64
• Factors of 64 are: 1, 2, 4, 8, 16, 32, 64.

(vi) 98

• 1 * 98 = 98
• 2 * 49 = 98
• 7 * 14 = 98
• Factors of 98 are: 1, 2, 7, 14, 49, 98.

Question 3.
Write the first six multiples of :
(i) 4
(ii) 9
(iii) 11
(iv) 15
(v) 18
(vi) 16

Solution :

(i) 4

• $4 \times 1 = 4$
• $4 \times 2 = 8$
• $4 \times 3 = 12$
• $4 \times 4 = 16$
• $4 \times 5 = 20$
• $4 \times 6 = 24$
• First six multiples of 4 are: 4, 8, 12, 16, 20, 24.

(ii) 9

• 9 * 1 = 9$
• 9 \times 2 = 18$
• 9 \times 3 = 27$
• 9 \times 4 = 36$
• 9 \times 5 = 45$
• 9 \times 6 = 54$
• First six multiples of 9 are: 9, 18, 27, 36, 45, 54.

(iii) 11

• 11 * 1 = 11
• 11 * 2 = 22
• 11 * 3 = 33
• 11 * 4 = 44
• 11 * 5 = 55
• 11 * 6 = 66
• First six multiples of 11 are: 11, 22, 33, 44, 55, 66.

(iv) 15

• 15 * 1 = 15
• 15 * 2 = 30
• 15 * 3 = 45
• 15 * 4 = 60
• 15 * 5 = 75
• 15 * 6 = 90
• First six multiples of 15 are: 15, 30, 45, 60, 75, 90.

(v) 18

• 18 * 1 = 18
• 18 * 2 = 36
• 18 * 3 = 54
• 18 * 4 = 72
• 18 * 5 = 90
• 18 * 6 = 108
• First six multiples of 18 are: 18, 36, 54, 72, 90, 108.

(vi) 16

• 16 * 1 = 16
• 16 * 2 = 32
• 16 * 3 = 48
• 16 * 4 = 64
• 16 * 5 = 80
• 16 * 6 = 96
• First six multiples of 16 are: 16, 32, 48, 64, 80, 96.

Question 4.
The product of two numbers is 36 and their sum is 13. Find the numbers.

Solution :

1. List the given conditions:

• Condition 1: The product of the two numbers must be 36.
• Condition 2: The sum of the same two numbers must be 13.

2. Find all factor pairs of 36:

We look for pairs of numbers that multiply to give 36:

• 1 * 36 = 36
• 2 * 18 = 36
• 3 * 12 = 36
• 4 * 9 = 36
• 6 * 6 = 36

3. Check the sum of each pair:

Now, we add the numbers in each pair to see which one equals 13:

• 1 + 36 = 37 (No)
• 2 + 18 = 20 (No)
• 3 + 12 = 15 (No)
• 4 + 9 = 13 (Yes!)
• 6 + 6 = 12 (No)

4. Conclusion:

The two numbers are 4 and 9.

Question 5.
The product of two numbers is 48 and their sum is 16. Find the numbers.

Solution :

1. Identify the Conditions:

• Condition 1: The product (multiplication) of the two numbers must be 48.
• Condition 2: The sum (addition) of those same two numbers must be 16.

2. List all Factor Pairs of 48:

Let’s find all pairs of numbers that multiply to give 48:

• 1 * 48 = 48
• 2 * 24 = 48
• 3 * 16 = 48
• 4 * 12 = 48
• 6 * 8 = 48

3. Check the Sum of each pair:

Now, we add the numbers in each pair to see which one equals 16:

• 1 + 48 = 49 (No)
• 2 + 24 = 26 (No)
• 3 + 16 = 19 (No)
• 4 + 12 = 16 (Yes!)
• 6 + 8 = 14 (No)

4. Conclusion:

The two numbers are 4 and 12.

Question 6.
Write two numbers which differ by 3 and whose product is 54.

Solution :

1. Identify the Conditions:

• Condition 1: The product (multiplication) must be 54.
• Condition 2: The difference (subtraction) must be 3.

2. List all Factor Pairs of 54:

Let’s find pairs of numbers that multiply to give 54:

• 1 * 54 = 54
• 2 * 27 = 54
• 3 * 18 = 54
• 6 * 9 = 54

3. Check the Difference of each pair:

Now, we subtract the smaller number from the larger number in each pair to find which one has a difference of 3:

• 54 – 1 = 53 (No)
• 27 – 2 = 25 (No)
• 18 – 3 = 15 (No)
• 9 – 6 = 3 (Yes!)

4. Conclusion:

The two numbers are 6 and 9.

Question 7.
Without making any actual division show that 7007 is divisible by 7.

Solution :

We can break 7007 into two parts that are clearly multiples of 7.

1. Break down 7007:

7007 = 7000 + 7

2. Check each part:

• We know that 7000 is 7 * 1000 (Divisible by 7)
• We know that 7 is 7 * 1 (Divisible by 7)

3. Conclusion:

Since both parts of the sum are divisible by 7, the entire number 7007 must be divisible by 7.

Question 8.
Without making any actual division, show that 2300023 is divisible by 23.

1. Expand the number 2,300,023:

We can write this number by separating the “23” parts:

2,300,023 = 2,300,000 + 23

2. Analyze the first part (2,300,000):

• 2,300,000 can be written as 23 * 100,000.
• Since it is 23 multiplied by a whole number, 2,300,000 is divisible by 23.

3. Analyze the second part (23):

• 23 can be written as 23 * 1.
• Since it is 23 multiplied by a whole number, 23 is divisible by 23.

4. Apply the Property of Divisibility:

The rule states: If a number is the sum of two numbers that are both divisible by $x$, then the whole number is also divisible by x.

• Part A (2,300,000) is divisible by 23.
• Part B (23) is divisible by 23.

Conclusion:

Since both parts are divisible by 23, the sum 2,300,023 is divisible by 23.

Question 9.
Without making any actual division, show that each of the following numbers is divisible
by 11.
(i) 11011
(ii) 110011
(iii) 11000011

Solution :

To show that these numbers are divisible by 11 without long division, we can use the Expansion Method by breaking each number into parts that are clearly multiples of 11.

Step-by-Step Solutions

(i) 11011

1. Expand the number: 11011 = 11000 + 11
2. Analyze the parts:
   • 11000 is 11 * 1000 (Divisible by 11)
   • 11 is 11 * 1 (Divisible by 11)
3. Conclusion: Since both parts are divisible by 11, the sum 11011 is divisible by 11.

(ii) 110011

1. Expand the number: 110011 = 110000 + 11
2. Analyze the parts:
   • 110000 is 11 * 10000 (Divisible by 11)
   • 11 is 11 * 1 (Divisible by 11)
3. Conclusion: Since both parts are divisible by 11, the sum 110011 is divisible by 11.

(iii) 11000011

1. Expand the number: 11000011 = 11000000 + 11
2. Analyze the parts:
   • 11000000 is 11 * 1000000 (Divisible by 11)
   • 11 is 11 * 1 (Divisible by 11)
3. Conclusion: Since both parts are divisible by 11, the sum 11000011 is divisible by 11.

Question 10.
Without actual division, show that each of the following numbers is divisible by 8 :
(i) 1608
(ii) 56008
(iii) 240008

Solution :

(i) 1608

1. Expand the number: 1600 + 8
2. Analyze the parts:
   • 1600 is 8 * 200 (Divisible by 8)
   • 8 is 8 * 1 (Divisible by 8)
3. Conclusion: Since both parts are divisible by 8, the sum 1608 is divisible by 8.

(ii) 56008

1. Expand the number: 56000 + 8
2. Analyze the parts:
   • 56000 is 8 * 7000
   • (Divisible by 8)
   • 8 is 8 * 1 (Divisible by 8)
3. Conclusion: Since both parts are divisible by 8, the sum 56008 is divisible by 8.

(iii) 240008

1. Expand the number: 240000 + 8
2. Analyze the parts:
   • 240000 is 8 * 30000 (Divisible by 8)
   • 8 is 8 *s 1 (Divisible by 8)
3. Conclusion: Since both parts are divisible by 8, the sum 240008 is divisible by 8.

EXERCISE 9(C)

Question 1.
find which of the following numbers are divisible by 2 :
(i) 352
(ii) 523
(iii) 496
(iv) 649

Solution :

(i) 352

• Last digit: 2
• Check: 2 is an even number.
• Conclusion: 352 is divisible by 2.

(ii) 523

• Last digit: 3
• Check: 3 is an odd number.
• Conclusion: 523 is not divisible by 2.

(iii) 496

• Last digit: 6
• Check: 6 is an even number.
• Conclusion: 496 is divisible by 2.

(iv) 649

• Last digit: 9
• Check: 9 is an odd number.
• Conclusion: 649 is not divisible by 2.

Question 2.
Find which of the following number are divisible by 4 :
(i) 222
(ii) 532
(iii) 678
(iv) 9232

Solution :

(i) 222

• Last two digits: 22
• Check: 22 / 4 = 5 with a remainder of 2.
• Conclusion: 222 is not divisible by 4.

(ii) 532

• Last two digits: 32
• Check: 32 / 4 = 8 (No remainder).
• Conclusion: 532 is divisible by 4.

(iii) 678

• Last two digits: 78
• Check: 78 / 4 = 19 with a remainder of 2.
• Conclusion: 678 is not divisible by 4.

(iv) 9232

• Last two digits: 32
• Check: 32 / 4 = 8 (No remainder).
• Conclusion: 9232 is divisible by 4.

Question 3.
Find the which of the following numbers are divisible by 8 :
(i) 324
(ii) 2536
(iii) 92760
(iv) 444320

Solution :

(i) 324

• Last three digits: 324
• Check: 324 / 8
  • 32 / 8 = 4
  • 4 / 8 leaves a remainder of 4.
• Conclusion: 324 is not divisible by 8.

(ii) 2536

• Last three digits: 536
• Check: 536 / 8
  • 8 / 60 = 480
  • 536 – 480 = 56
  • 56 / 8 = 7 (Total: 67, no remainder).
• Conclusion: 2536 is divisible by 8.

(iii) 92760

• Last three digits: 760
• Check: 760 / 8
  • 720 / 8 = 90
  • 40 / 8 = 5 (Total: 95, no remainder).
• Conclusion: 92760 is divisible by 8.

(iv) 444320

• Last three digits: 320
• Check: 320 / 8
  • 32 / 8 = 4, so 320 / 8 = 40 (No remainder).
• Conclusion: 444320 is divisible by 8.

Question 4.
Find which of the following numbers are divisible by 3 :
(i) 221
(ii) 543
(iii) 28492
(iv) 92349

Solution :

(i) 221

• Sum of digits: 2 + 2 + 1 = 5
• Check: 5 is not divisible by 3.
• Conclusion: 221 is not divisible by 3.

(ii) 543

• Sum of digits: 5 + 4 + 3 = 12
• Check: 12 / 3 = 4 (No remainder).
• Conclusion: 543 is divisible by 3.

(iii) 28492

• Sum of digits: 2 + 8 + 4 + 9 + 2 = 25
• Check: 25 is not divisible by 3 (3 \times 8 = 24, remainder 1).
• Conclusion: 28492 is not divisible by 3.

(iv) 92349

• Sum of digits: 9 + 2 + 3 + 4 + 9 = 27
• Check: 27 / 3 = 9 (No remainder).
• Conclusion: 92349 is divisible by 3.

Question 5.
Find which of the following numbers are divisible by 9 :
(i) 1332
(ii) 53247
(iii) 4968
(iv) 200314

Solution :

(i) 1332

• Sum of digits: 1 + 3 + 3 + 2 = 9
• Check: 9 is divisible by 9 (9 * 1 = 9).
• Conclusion: 1332 is divisible by 9.

(ii) 53247

• Sum of digits: 5 + 3 + 2 + 4 + 7 = 21
• Check: 21 is not divisible by 9 (9 * 2 = 18, remainder 3).
• Conclusion: 53247 is not divisible by 9.

(iii) 4968

• Sum of digits: 4 + 9 + 6 + 8 = 27
• Check: 27 is divisible by 9 (9 * 3 = 27).
• Conclusion: 4968 is divisible by 9.

(iv) 200314

• Sum of digits: 2 + 0 + 0 + 3 + 1 + 4 = 10
• Check: 10 is not divisible by 9.
• Conclusion: 200314 is not divisible by 9.

Question 6.
Find which of the following number are divisible by 6 :
(i) 324
(ii) 2010
(iii) 33278
(iv) 15505

Solution :

(i) 324

• Divisible by 2? Yes, the last digit is 4 (even).
• Divisible by 3? Sum of digits: 3 + 2 + 4 = 9$. Since 9 is divisible by 3, yes.
• Conclusion: 324 is divisible by 6.

(ii) 2010

• Divisible by 2? Yes, the last digit is 0 (even).
• Divisible by 3? Sum of digits: 2 + 0 + 1 + 0 = 3. Since 3 is divisible by 3, yes.
• Conclusion: 2010 is divisible by 6.

(iii) 33278

• Divisible by 2? Yes, the last digit is 8 (even).
• Divisible by 3? Sum of digits: 3 + 3 + 2 + 7 + 8 = 23. 23 is not divisible by 3.
• Conclusion: 33278 is not divisible by 6.

(iv) 15505

• Divisible by 2? No, the last digit is 5 (odd).
• Divisible by 3? (Not needed, but 1+5+5+0+5 = 16, no).
• Conclusion: 15505 is not divisible by 6.

Question 7.
Find which of the following numbers are divisible by 5 :
(i) 5080
(ii) 66666
(iii) 755
(iv) 9207

Solution :

(i) 5080

• Last digit: 0
• Check: Since it ends in 0, it satisfies the rule.
• Conclusion: 5080 is divisible by 5.

(ii) 66666

• Last digit: 6
• Check: 6 is neither 0 nor 5.
• Conclusion: 66666 is not divisible by 5.

(iii) 755

• Last digit: 5
• Check: Since it ends in 5, it satisfies the rule.
• Conclusion: 755 is divisible by 5.

(iv) 9207

• Last digit: 7
• Check: 7 is neither 0 nor 5.
• Conclusion: 9207 is not divisible by 5.

Question 8.
Find which of the following numbers are divisible by 10 :
(i) 9990
(ii) 0
(iii) 847
(iv) 8976

Solution :

(i) 9990

• Last digit: 0
• Check: Since the last digit is 0, it satisfies the rule.
• Conclusion: 9990 is divisible by 10.

(ii) 0

• Last digit: 0
• Check: Zero is divisible by every non-zero integer ($0 \div 10 = 0$).
• Conclusion: 0 is divisible by 10.

(iii) 847

• Last digit: 7
• Check: 7 is not 0.
• Conclusion: 847 is not divisible by 10.

(iv) 8976

• Last digit: 6
• Check: 6 is not 0.
• Conclusion: 8976 is not divisible by 10.

Question 9.
Find which of the following numbers are divisible by 11 :
(i) 5918
(ii) 68,717
(iii) 3882

(iv) 10857

Solution :

(i) 5918

• Odd places (8, 9): 8 + 9 = 17
• Even places (1, 5): 1 + 5 = 6
• Difference: 17 – 6 = 11
• Conclusion: 11 is a multiple of 11. 5918 is divisible by 11.

(ii) 68,717

• Odd places (7, 7, 6): 7 + 7 + 6 = 20
• Even places (1, 8): 1 + 8 = 9
• Difference: 20 – 9 = 11
• Conclusion: 11 is a multiple of 11. 68,717 is divisible by 11.

(iii) 3882

• Odd places (2, 8): 2 + 8 = 10
• Even places (8, 3): 8 + 3 = 11
• Difference: 11 – 10 = 1
• Conclusion: 1 is not 0 or a multiple of 11. 3882 is not divisible by 11.

(iv) 10857

• Odd places (7, 8, 1): 7 + 8 + 1 = 16
• Even places (5, 0): 5 + 0 = 5
• Difference: 16 – 5 = 11
• Conclusion: 11 is a multiple of 11. 10857 is divisible by 11.

Question 10.
Find which of the following numbers are divisible by 15 :
(i) 960
(ii) 8295
(iii) 10243
(iv) 5013

Solution :

(i) 960

• Divisible by 5? Yes, the last digit is 0.
• Divisible by 3? Sum of digits: 9 + 6 + 0 = 15. Since 15 is divisible by 3, yes.
• Conclusion: Since it is divisible by both 3 and 5, 960 is divisible by 15.

(ii) 8295

• Divisible by 5? Yes, the last digit is 5.
• Divisible by 3? Sum of digits: 8 + 2 + 9 + 5 = 24. Since 24 is divisible by 3, yes.
• Conclusion: Since it is divisible by both 3 and 5, 8295 is divisible by 15.

(iii) 10243

• Divisible by 5? No, the last digit is 3 (not 0 or 5).
• Divisible by 3? Sum of digits: 1 + 0 + 2 + 4 + 3 = 10$le by 15.

(iv) 5013

• Divisible by 5? No, the last digit is 3 (not 0 or 5).
• Divisible by 3? Sum of digits: 5 + 0 + 1 + 3 = 9. Divisible by 3.
• Conclusion: Even though it is divisible by 3, it is not divisible by 5, so 5013 is not divisible by 15.

Question 11.
In each of the following numbers, replace M by the smallest number to make resulting
number divisible by 3 :
(i) 64 M 3
(ii) 46 M 46
(iii) 27 M 53

Solution :

(i) 64 M 3

• Sum of known digits: 6 + 4 + 3 = 13
• Condition: 13 + M must be a multiple of 3.
• Multiples of 3 after 13: 15, 18, 21…
• Smallest value for M: To get the sum to 15, we need 13 + M = 15, which means M = 2.
• Result: M = 2

(ii) 46 M 46

• Sum of known digits: 4 + 6 + 4 + 6 = 20
• Condition: 20 + M must be a multiple of 3.
• Multiples of 3 after 20: 21, 24, 27…
• Smallest value for M: To get the sum to 21, we need 20 + M = 21, which means M = 1.
• Result: M = 1

(iii) 27 M 53

• Sum of known digits: 2 + 7 + 5 + 3 = 17
• Condition: 17 + M must be a multiple of 3.
• Multiples of 3 after 17: 18, 21, 24…
• Smallest value for M: To get the sum to 18, we need 17 + M = 18, which means M = 1.
• Result: M = 1

Question 12.
In each of the following numbers replace M by the smallest number to make resulting
number divisible by 9.
(i) 76 M 91
(ii) 77548 M
(iii) 627 M 9

Solution :

(i) 76 M 91

• Sum of known digits: 7 + 6 + 9 + 1 = 23
• Condition: 23 + M must be a multiple of 9.
• Multiples of 9 after 23: 27, 36…
• Smallest value for M: To get the sum to 27, we need 23 + M = 27, which means M = 4.
• Result: M = 4

(ii) 77548 M

• Sum of known digits: 7 + 7 + 5 + 4 + 8 = 31
• Condition: 31 + M must be a multiple of 9.
• Multiples of 9 after 31: 36, 45…
• Smallest value for M: To get the sum to 36, we need 31 + M = 36, which means M = 5.
• Result: M = 5

(iii) 627 M 9

• Sum of known digits: 6 + 2 + 7 + 9 = 24
• Condition: 24 + M must be a multiple of 9.
• Multiples of 9 after 24: 27, 36…
• Smallest value for M: To get the sum to 27, we need 24 + M = 27, which means M = 3.
• Result: M = 3

Question 13.
In each of the following numbers, replace M by the smallest number to make resulting
number divisible by 11.
(i) 39 M 2
(ii) 3 M 422
(iii) 70975 M
(iv) 14 M 75

Solution :

(i) 39M2

• Odd places (2, 9): 2 + 9 = 11
• Even places (M, 3): M + 3
• Difference: (M + 3) – 11 = M – 8
• Condition: For the difference to be 0, M – 8 = 0, so M = 8.
• Result: M = 8

(ii) 3 M 422

• Odd places (2, 4, 3): 2 + 4 + 3 = 9
• Even places (2, M): 2 + M
• Difference: 9 – (2 + M) = 7 – M
• Condition: For the difference to be 0, 7 – M = 0, so M = 7.
• Result: M = 7

(iii) 70975 M

• Odd places (M, 7, 0): M + 7 + 0 = M + 7
• Even places (5, 9, 7): 5 + 9 + 7 = 21
• Difference: 21 – (M + 7) = 14 – M
• Condition: If M = 3, 14 – 3 = 11. Since 11 is a multiple of 11, M = 3 works.
• Result: M = 3

(iv) 14 M 75

• Odd places (5, M, 1): 5 + M + 1 = M + 6
• Even places (7, 4): 7 + 4 = 11
• Difference: (M + 6) – 11 = M – 5
• Condition: For the difference to be 0, M – 5 = 0, so M = 5.
• Result: M = 5

Question 14.
State, true or false :
(i) If a number is divisible by 4. It is divisible by 8.

(ii) If a number is a factor of 16 and 24, it is a factor of 48.
(iii) If a number is divisible by 18, it is divisible by 3 and 6.
(iv) If a divide b and c completely, then a divides (i) a + b (ii) a – b also completely.

Solution :

(i) False
(ii) True
(iii) True
(iv) True