Page 108 & 110
Divisibility Rules of 2, 5, 10 and 4 - Do these
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1) Circle the numbers which are divisible by 4. Give the reason, if it is not divisible by 4.
Numbers: 2436, 3840, 1235, 3636, 6850, 5644, 8888, 6430.Numbers divisible by 4: 2436, 3840, 3636, 5644, 8888 (their last two digits are divisible by 4).
Reasons for not divisible by 4:
- 1235: Last two digits '35' is not divisible by 4.
- 6850: Last two digits '50' is not divisible by 4.
- 6430: Last two digits '30' is not divisible by 4. -
2) Write the missing number in the blank to make the number exactly divisible by 4.
a) 3232 (or 3236)
b) 3040 (or 3044, 3048)
c) 5816 (or 5836, 5856, 5876, 5896)
d) 532 (or 536)
e) 6500 (or 6504, 6508, 6512...)
Page 112
Divisibility Rule of 8 - Do these
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Circle the following numbers which are divisible by 8: 42456, 73791, 68240, 93230, 27000.
Numbers divisible by 8: 42456, 68240, 27000
Explanation:
- 42456: Last 3 digits 456 ÷ 8 = 57 (Divisible)
- 73791: Last 3 digits 791 is odd (Not divisible)
- 68240: Last 3 digits 240 ÷ 8 = 30 (Divisible)
- 93230: Last 3 digits 230 ÷ 8 = 28.75 (Not divisible)
- 27000: Last 3 digits 000 (Divisible by 8)
Page 114
EXERCISE-1
Questions 1 to 6
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1. Circle the numbers which are divisible by 2 (by using divisibility rule): 3624, 3549, 7864, 8420, 8500, 8646.
Numbers divisible by 2: 3624, 7864, 8420, 8500, 8646 (ending in 0, 4, 6).
- 2. 480 (or 484, 488) - Fill in the blank with a suitable digit to make the number divisible by 4.
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3. Fill in the blank with a suitable digit to make the number divisible by both 2 and 10.
6780, 5880, 3880, 2220, 3640, 7860 (Must end in 0). -
4. Identify the numbers which are divisible by both 4 and 8: 2104, 726352, 1800, 32256, 52248, 25608.
Numbers divisible by both 4 and 8: 2104, 726352, 32256, 52248, 25608
(Note: 1800 is divisible by 4 but not by 8 because 800 ÷ 8 = 100, wait: 1800 ÷ 8 = 225, so 1800 is also divisible by both 4 and 8). -
5. Fill the missing digit that would make each number divisible by the number given:
a) 3950 by 10
b) 20712 by 4 (or 20716)
c) 92048 by 2 (Already ends in 8, digit is 8)
d) 1456 by 8 (56 ÷ 8 = 7)
e) 23400 (or 5) by 5 -
6. Find the smallest number to be added to 2887, so that it can be divisible by 4.
2887 ÷ 4 leaves remainder 3.
Smallest number to add = 4 - 3 = 1 (2887 + 1 = 2888, which is divisible by 4).
Page 116
Common Multiples - Do these
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Write the first 10 multiples of the following numbers and list the common multiples.
a) 2 and 4Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Common multiples: 4, 8, 12, 16, 20 -
b) 4 and 12
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Common multiples: 12, 24, 36 -
c) 6 and 8
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
Common multiples: 24, 48 -
d) 5 and 10
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Common multiples: 10, 20, 30, 40, 50
Page 118
Least Common Multiple (L.C.M) - Do these
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1. Find the L.C.M for the following sets of numbers:
a) 12, 15 → L.C.M = 60
b) 16, 20 → L.C.M = 80
c) 8, 12, 20 → L.C.M = 120
d) 15, 20 → L.C.M = 60
e) 6, 9, 12 → L.C.M = 36 -
2. Find the L.C.M for the following numbers. What do you observe?
a) 15, 30 → L.C.M = 30
b) 4, 16 → L.C.M = 16
c) 5, 15 → L.C.M = 15
d) 6, 18 → L.C.M = 18Observation: In a given pair of numbers, if one of them is a multiple of the other, then the larger number is the L.C.M of the pair.
Page 120 & 122
Factors Activity
- Factors of 10: 1 × 10 = 10, 2 × 5 = 10. Factors are: 1, 2, 5, 10
- Factors of 14: 1 × 14 = 14, 2 × 7 = 14. Factors are: 1, 2, 7, 14
Page 124
Factors & Prime Numbers - Do these
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1) Find all the factors of the following numbers.
a) 21 : 1, 3, 7, 21
b) 38 : 1, 2, 19, 38 -
2) Find out whether the first number is a factor of the second number.
a) 14; 322 → Yes (322 ÷ 14 = 23)
b) 16; 832 → Yes (832 ÷ 16 = 52)
c) 5; 425 → Yes (425 ÷ 5 = 85)
d) 25; 3500 → Yes (3500 ÷ 25 = 140)
e) 8; 48 → Yes (48 ÷ 8 = 6)
f) 14; 37 → No (37 ÷ 14 has remainder)
g) 15; 75 → Yes (75 ÷ 15 = 5)
h) 12; 72 → Yes (72 ÷ 12 = 6) -
3) Find the odd factors of 22.
Factors of 22 = 1, 2, 11, 22. Odd factors = 1, 11
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4) Write all the even factors of 34.
Factors of 34 = 1, 2, 17, 34. Even factors = 2, 34
- 5) List out the numbers, which are prime/composite below 30.
| Prime Number | Composite Number |
|---|---|
| 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 | 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28 |
Page 126
Sieve of Eratosthenes Table
| Numbers | 1-10 | 10-20 | 20-50 | 50-100 |
|---|---|---|---|---|
| Prime numbers | 2, 3, 5, 7 | 11, 13, 17, 19 | 23, 29, 31, 37, 41, 43, 47 | 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
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1. Write the total prime numbers from 1 to 100?
Total prime numbers = 25
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2. Are all the prime numbers even or odd?
All prime numbers except '2' are odd. '2' is the only even prime number.
Page 130
Prime Factorisation - Do these
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1) Write the prime-factorisation for the following numbers.
a) 52 = 2 × 2 × 13
b) 100 = 2 × 2 × 5 × 5
c) 88 = 2 × 2 × 2 × 11
d) 90 = 2 × 3 × 3 × 5 -
2) The prime factorisation of 12 × 15 is:
12 = 2 × 2 × 3, 15 = 3 × 5
12 × 15 = 2 × 2 × 3 × 3 × 5
3) Match the following
| A | Ans | B |
|---|---|---|
| a) 2 × 2 × 2 × 3 × 3 × 5 | ( 2 ) | 1) 180 |
| b) 2 × 2 × 2 × 3 × 5 × 5 | ( 4 ) | 2) 360 |
| c) 2 × 2 × 3 × 3 × 5 | ( 1 ) | 3) 900 |
| d) 2 × 3 × 3 × 5 × 5 | ( 5 ) | 4) 600 |
| e) 2 × 2 × 3 × 3 × 5 × 5 | ( 3 ) | 5) 450 |
- 4) 5 × 2 × 3 × 3 is the prime factorisation of 90
Page 132 & 134
Highest Common Factor (H.C.F) - Do these
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Find the H.C.F of the following pairs of numbers by using common factors method:
1) 21 and 28 : Factors of 21 = 1, 3, 7, 21; Factors of 28 = 1, 2, 4, 7, 14, 28 → H.C.F = 7
2) 34 and 20 : Factors of 34 = 1, 2, 17, 34; Factors of 20 = 1, 2, 4, 5, 10, 20 → H.C.F = 2
3) 33 and 39 : Factors of 33 = 1, 3, 11, 33; Factors of 39 = 1, 3, 13, 39 → H.C.F = 3
4) 16 and 36 : Factors of 16 = 1, 2, 4, 8, 16; Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 → H.C.F = 4
5) 12 and 18 : Factors of 12 = 1, 2, 3, 4, 6, 12; Factors of 18 = 1, 2, 3, 6, 9, 18 → H.C.F = 6
6) 80 and 100 : Factors of 80 = 1, 2, 4, 5, 8, 10, 16, 20, 40, 80; Factors of 100 = 1, 2, 4, 5, 10, 20, 25, 50, 100 → H.C.F = 20
Page 138
L.C.M and H.C.F by Methods - Do these
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1) Find L.C.M and H.C.F by prime factorisation method:
a) 15, 48 → 15 = 3 × 5, 48 = 2 × 2 × 2 × 2 × 3
H.C.F = 3, L.C.M = 240 -
b) 18, 48 → 18 = 2 × 3 × 3, 48 = 2 × 2 × 2 × 2 × 3
H.C.F = 6, L.C.M = 144 -
c) 15, 25 → 15 = 3 × 5, 25 = 5 × 5
H.C.F = 5, L.C.M = 75 -
2) Find L.C.M and H.C.F by division method:
a) 28, 36 → H.C.F = 4, L.C.M = 252
b) 12, 18 → H.C.F = 6, L.C.M = 36
c) 30, 90 → H.C.F = 30, L.C.M = 90
Page 140
Product of two numbers = H.C.F × L.C.M Table
| S.No | First number | Second number | Product of two numbers | H.C.F | L.C.M | H.C.F × L.C.M |
|---|---|---|---|---|---|---|
| 1 | 9 | 12 | 108 | 3 | 36 | 108 |
| 2 | 15 | 20 | 300 | 5 | 60 | 300 |
| 3 | 18 | 15 | 270 | 3 | 90 | 270 |
| 4 | 8 | 12 | 96 | 4 | 24 | 96 |
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2. The H.C.F of two numbers is 3 and their L.C.M is 90. If one number is 15, find the other number.
Other number = (H.C.F × L.C.M) ÷ First number = (3 × 90) ÷ 15 = 270 ÷ 15 = 18
Page 142 & 144
EXERCISE-2
Real-Life Problems on L.C.M and H.C.F
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1) The H.C.F of two numbers is 6 and their L.C.M is 72. If one number is 18, find the other number.
Other number = (6 × 72) ÷ 18 = 432 ÷ 18 = 24
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2) The H.C.F of two numbers is 6. Their L.C.M is 36. If one number is 12, find the other number.
Other number = (6 × 36) ÷ 12 = 216 ÷ 12 = 18
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3) Sita exercises every 6th day and Gita exercises every 8th day. Today they both exercised. After how many days will they exercise together again?
L.C.M of 6 and 8 = 24 days.
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4) Ramu has 16 blue marbles and 12 black marbles. If he wants to arrange them in identical groups without leaving any marbles, what is the maximum number of marbles in each group Ramu can make?
H.C.F of 16 and 12 = 4 marbles.
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5) Two Neon lights are turned on at the same time. One blinks for every 4 seconds and another blinks for every 6 seconds. How many times will they blink together in one minute? (1 minute = 60 seconds)
L.C.M of 4 and 6 = 12 seconds.
In 60 seconds, they blink together: 60 ÷ 12 = 5 times. -
6) There are 40 girls and 32 boys who want to participate in state-level games competition. Each team must have the same number of girls and the same number of boys.
a) How many boys and girls will be there in each team?
b) What is the maximum number of students in each team that can participate?Number of teams = H.C.F of 40 and 32 = 8 teams.
a) In each team: 40 ÷ 8 = 5 girls and 32 ÷ 8 = 4 boys.
b) Total students in each team = 5 + 4 = 9 students. -
7) What is the least number of chairs needed for an auditorium so that they can be arranged either 27 or 33 in a row?
L.C.M of 27 and 33 = 297 chairs.
-
8) A number which has more than two factors is
Answer: D) composite number
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9) Which of the following number is not a factor of 56?
Answer: B) 6
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10) Which of the following number is exactly divisible by all 2, 5 and 10?
Answer: B) 360
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11) A number is divisible by 8 if its last ________ digits are divisible by 8.
Answer: B) 3
Page 146
Improve Your Learning
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1. 5 and 7 are common factors of 35 and ________.
Answer: c) 70 (35 × 2 = 70)
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2. Which of the following numbers is NOT a factor of 56?
Answer: b) 6
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3. The factors of 28 include 1, 2, 4, and 28. Find the remaining two factors.
Remaining factors: 7 and 14
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4. Write the first three multiples of 16?
16, 32, 48
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5. Two wires with the length of 56 m and 72 m are cut into small pieces of equal length. What is the maximum possible length of each piece?
H.C.F of 56 and 72 = 8 meters.
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6. What numbers should be written in the shaded part (common multiples of 2 and 5)?
Numbers in shaded part (multiples of 10): 10, 20, 30
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7. A student says, "If two numbers are even, their HCF must be at least 2." Is this student correct? Explain your reasoning and provide an example.
Yes, the student is correct.
Reasoning: Every even number is divisible by 2. Therefore, 2 is always a common factor of any two even numbers, making their HCF at least 2.
Example: For 6 and 8, common factors are 1 and 2, so HCF is 2. -
8. Vani took some marbles. When she makes groups of 5, 4 or 6, each time one marble is leftover. What is the smallest number of marbles that Vani had?
L.C.M of 4, 5, and 6 = 60.
Smallest number of marbles = 60 + 1 = 61 marbles. -
9. Amit sets an alarm after every 1 hour (60 min). Sumit sets an alarm after every 45 minutes. If they start studying at the same time, after how many hours will their alarms ring together?
L.C.M of 60 minutes and 45 minutes = 180 minutes.
180 minutes ÷ 60 = 3 hours. -
10. Anu visits the library once in 6 days and Shreya visits once in 4 days. If they both visit on a Monday, on which day will they visit the library together the next time?
L.C.M of 6 and 4 = 12 days.
Starting from Monday, adding 12 days gives: Monday + 12 days = Saturday.
