HCF is the greatest number that divides two or more numbers exactly. LCM is the smallest number that is a multiple of two or more numbers. Important questions class 7 maths chapter 11 cover Finding Common Ground, common factors, common multiples, prime factorisation, HCF, LCM, and real-life word problems.
A square tile that fits a room perfectly and equal-weight bags for rice both need the same idea: common factors. Class 7 Maths Chapter 11, Finding Common Ground, helps students decide when to use HCF and when to use LCM. This chapter becomes easier when students connect every problem to one question: are we finding the largest equal division or the smallest common meeting point?
Key Takeaways
| Concept |
What You Need to Know |
| Common Factors |
Factors shared by two or more numbers |
| HCF / GCD |
Highest among all common factors |
| Prime Factorisation |
Writing a number as a product of primes |
| Division Method |
Prime factorisation using repeated division |
| Common Multiples |
Multiples shared by two or more numbers |
| LCM |
Lowest among all common multiples |
| HCF Rule |
Take common primes with minimum powers |
| LCM Rule |
Take all primes with maximum powers |
| Key Property |
HCF × LCM = Product of two numbers |
| Word Problems |
HCF for largest equal groups, LCM for first common event |
Class 7 Maths Chapter List
Important Topics in Class 7 Maths Chapter 11
Chapter 11 maths class 7 important questions mostly come from HCF, LCM, prime factorisation, and word problems.
- Common factors
- Highest Common Factor or HCF
- Greatest Common Divisor or GCD
- Prime numbers
- Prime factorisation
- Division method
- HCF using prime factorisation
- Common multiples
- Lowest Common Multiple or LCM
- LCM using prime factorisation
- HCF × LCM property
- HCF and LCM word problems
- Efficient combined procedure
Very Short Answer Questions: Finding Common Ground Class 7 Important Questions
Very short answers test definitions and rules. Keep these answers direct.
Class 7 Maths Chapter 11 Important Questions
Q1. What is the HCF of two numbers?
Ans. The HCF is the highest number that divides both numbers exactly.
It is also called the GCD or Greatest Common Divisor.
Q2. What is the LCM of two numbers?
Ans. The LCM is the smallest number that is a multiple of both given numbers.
Q3. What is prime factorisation?
Ans. Prime factorisation means writing a number as a product of prime factors only.
Example: 84 = 2 × 2 × 3 × 7.
Q4. What is a common factor?
Ans. A common factor is a number that divides two or more numbers exactly.
Example: 4 is a common factor of 12 and 16.
Q5. What is a common multiple?
Ans. A common multiple is a number that is a multiple of two or more numbers.
Example: 24 is a common multiple of 6 and 8.
Q6. If one number is a factor of another, what is their HCF?
Ans. The HCF is the smaller number.
Example: HCF of 6 and 18 is 6.
Q7. If one number is a multiple of another, what is their LCM?
Ans. The LCM is the larger number.
Example: LCM of 3 and 24 is 24.
Q8. What is the HCF of two co-prime numbers?
Ans. The HCF of two co-prime numbers is 1.
Q9. What is the HCF of two consecutive numbers?
Ans. The HCF of two consecutive numbers is always 1.
Q10. State the relation between HCF, LCM, and product of two numbers.
Ans. HCF × LCM = Product of the two numbers.
Short Answer Questions on HCF and Common Factors
HCF questions ask for the largest number that divides all given numbers.
Use factor listing for small numbers and prime factorisation for larger numbers.
HCF and LCM Class 7 Questions and Answers
Q1. Find the common factors of 84 and 108.
Ans.
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12
Q2. Find the common factors of 14 and 30.
Ans.
Factors of 14: 1, 2, 7, 14
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Common factors: 1, 2
HCF = 2
Q3. Find the common factors of 28 and 42.
Ans.
Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Common factors: 1, 2, 7, 14
HCF = 14
Q4. Sameeksha’s room is 12 ft by 16 ft. She wants the largest square tiles that fit perfectly. What size tile should she buy?
Ans. The tile side must divide both 12 and 16 exactly.
Common factors of 12 and 16 are 1, 2, and 4.
Largest common factor = 4
So, each square tile should have side 4 ft.
Number of tiles = 12/4 × 16/4
= 3 × 4
= 12 tiles
Q5. Find the HCF of 50 and 60 using prime factorisation.
Ans.
50 = 2 × 5 × 5
60 = 2 × 2 × 3 × 5
Common primes are 2 and 5.
HCF = 2 × 5
= 10
Q6. Find the HCF of 77 and 725.
Ans.
77 = 7 × 11
725 = 5 × 5 × 29
There is no common prime factor.
HCF = 1
So, 77 and 725 are co-prime.
Prime Factorisation Class 7 Important Questions
Prime factorisation means breaking a number into prime numbers only.
Do not stop at composite factors like 4, 6, 9, or 21.
Prime Factorisation by Division Method
Q1. Find the prime factorisation of 105.
Ans.
105 = 3 × 35
35 = 5 × 7
So,
105 = 3 × 5 × 7
Q2. Find the prime factorisation of 225.
Ans.
225 = 5 × 45
45 = 5 × 9
9 = 3 × 3
So,
225 = 5 × 5 × 3 × 3
= 3² × 5²
Q3. Find the prime factorisation of 840.
Ans.
840 = 2 × 420
= 2 × 2 × 210
= 2 × 2 × 2 × 105
= 2 × 2 × 2 × 3 × 35
= 2 × 2 × 2 × 3 × 5 × 7
So,
840 = 2³ × 3 × 5 × 7
Q4. Find the prime factorisation of 1200.
Ans.
1200 = 2 × 600
= 2 × 2 × 300
= 2 × 2 × 2 × 150
= 2 × 2 × 2 × 2 × 75
= 2⁴ × 3 × 5²
Q5. List all factors of 225 using prime factorisation.
Ans.
225 = 3² × 5²
Factors: 1, 3, 5, 9, 15, 25, 45, 75, 225
Q6. Is 2 × 2 × 7 a factor of 840?
Ans.
2 × 2 × 7 = 28
840 ÷ 28 = 30
So, 28 is a factor of 840.
Q7. Is 3 × 3 × 3 a factor of 840?
Ans.
3 × 3 × 3 = 27
840 has only one factor of 3 in its prime factorisation.
So, 27 is not a factor of 840.
HCF and LCM Using Prime Factorisation
This is the most important calculation section in Finding Common Ground.
For HCF, take common primes with minimum powers. For LCM, take all primes with maximum powers.
Class 7 Maths Chapter 11 Important Questions with Answers
Q1. Find the HCF of 45 and 75.
Ans.
45 = 3 × 3 × 5
75 = 3 × 5 × 5
Common primes: 3 and 5
HCF = 3 × 5
= 15
Q2. Find the HCF of 112 and 84.
Ans.
112 = 2 × 2 × 2 × 2 × 7
84 = 2 × 2 × 3 × 7
Common primes: two 2s and one 7
HCF = 2 × 2 × 7
= 28
Q3. Find the HCF of 225 and 750.
Ans.
225 = 3 × 3 × 5 × 5
750 = 2 × 3 × 5 × 5 × 5
Common primes: one 3 and two 5s
HCF = 3 × 5 × 5
= 75
Q4. Find the HCF of 42, 75, and 24.
Ans.
42 = 2 × 3 × 7
75 = 3 × 5 × 5
24 = 2 × 2 × 2 × 3
Only common prime across all three numbers is 3.
HCF = 3
Q5. Find the LCM of 14 and 35.
Ans.
14 = 2 × 7
35 = 5 × 7
LCM = 2 × 5 × 7
= 70
Q6. Find the LCM of 96 and 360.
Ans.
96 = 2⁵ × 3
360 = 2³ × 3² × 5
LCM = 2⁵ × 3² × 5
= 32 × 9 × 5
= 1440
Q7. Find the LCM of 30 and 72.
Ans.
30 = 2 × 3 × 5
72 = 2³ × 3²
LCM = 2³ × 3² × 5
= 8 × 9 × 5
= 360
Q8. Find the LCM of 36 and 54.
Ans.
36 = 2² × 3²
54 = 2 × 3³
LCM = 2² × 3³
= 4 × 27
= 108
HCF and LCM Properties Class 7 Questions
Properties help students solve pattern-based questions faster.
These hcf and lcm properties class 7 questions test concept clarity, not only calculation.
Pattern-Based Questions from Finding Common Ground
Q1. Why is the HCF of 6 and 18 equal to 6?
Ans. 6 is a factor of 18.
When one number is a factor of the other, the HCF is the smaller number.
So, HCF of 6 and 18 is 6.
Q2. What is the HCF of two consecutive numbers?
Ans. The HCF of two consecutive numbers is always 1.
Example: 7 and 8 have no common factor except 1.
Q3. What is the HCF of two consecutive even numbers?
Ans. The HCF of two consecutive even numbers is always 2.
Example: HCF of 8 and 10 is 2.
Q4. What happens to the HCF if both numbers are doubled?
Ans. The HCF also doubles.
Example:
HCF of 270 and 50 is 10.
HCF of 540 and 100 is 20.
Q5. Why is the LCM of 3 and 24 equal to 24?
Ans. 24 is a multiple of 3.
When one number is a multiple of the other, the LCM is the larger number.
Q6. What is the LCM of two co-prime numbers?
Ans. The LCM of two co-prime numbers is their product.
Example:
LCM of 7 and 11 = 77
Q7. Verify HCF × LCM = Product for 105 and 95.
Ans.
105 = 3 × 5 × 7
95 = 5 × 19
HCF = 5
LCM = 3 × 5 × 7 × 19
= 1995
HCF × LCM = 5 × 1995
= 9975
Product = 105 × 95
= 9975
Verified.
Q8. Find two numbers whose HCF is 1 and LCM is 66.
Ans.
66 = 2 × 3 × 11
One valid pair is 6 and 11.
HCF of 6 and 11 = 1
LCM of 6 and 11 = 66
HCF LCM Word Problems Class 7
Word problems are the most important part of this chapter.
Use HCF for the largest equal group or biggest possible size. Use LCM for the first common time, shortest common length, or smallest number divisible by given numbers.
Real-Life Word Problems on HCF and LCM
Q1. Lekhana has 84 kg rice from one farm and 108 kg rice from another. She wants equal-weight bags using as few bags as possible. What should each bag weigh?
Ans. This is an HCF problem.
84 = 2 × 2 × 3 × 7
108 = 2 × 2 × 3 × 3 × 3
HCF = 2 × 2 × 3
= 12
Each bag should weigh 12 kg.
Number of bags = 84/12 + 108/12
= 7 + 9
= 16 bags
Q2. Anshu uses 6 cm cloth strips and Guna uses 8 cm strips. What is the shortest toran length both can make?
Ans. This is an LCM problem.
6 = 2 × 3
8 = 2 × 2 × 2
LCM = 2³ × 3
= 24
Both can make a 24 cm toran.
Q3. A sweet shop gives gajak every Monday. Kabamai visits every 10 days. Today is Monday. After how many days will she get free gajak again?
Ans. The shop gives gajak every 7 days.
Kabamai visits every 10 days.
We need LCM of 7 and 10.
LCM = 70
She will get free gajak again after 70 days.
Q4. A box has length 12 cm, width 18 cm, and height 36 cm. Which cube sizes fit without gaps?
Ans. Cube side must divide 12, 18, and 36.
HCF of 12, 18, and 36 is 6.
Possible cube sizes are factors of 6:
1 cm, 2 cm, 3 cm, and 6 cm
So, 1 cm, 2 cm, 3 cm, and 6 cm cubes fit without gaps.
Q5. A cowherd’s cows pass equally through gates of 3, 5, and 7. The total is less than 200. How many cows are there?
Ans. The number must be divisible by 3, 5, and 7.
LCM of 3, 5, and 7 = 105
The only multiple of 105 less than 200 is 105.
So, there are 105 cows.
Q6. Find the smallest number divisible by 3, 4, 5, and 7 that leaves remainder 10 when divided by 11.
Ans.
LCM of 3, 4, 5, and 7 = 420
Multiples of 420 are 420, 840, 1260, 1680, 2100.
Now check division by 11:
2100 ÷ 11 leaves remainder 10.
So, the required number is 2100.
Q7. What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, and 10?
Ans.
LCM = 2³ × 3² × 5
= 8 × 9 × 5
= 360
The smallest number is 360.
Efficient Procedure for Finding HCF and LCM Together
The efficient procedure finds both HCF and LCM in one table.
Divide both numbers by common prime factors until no common factor remains. HCF is the product of common divisors. LCM is the product of common divisors and remaining numbers.
Class 7 Maths Chapter 11 Extra Questions
Q1. Find the HCF and LCM of 84 and 180 using the efficient procedure.
Ans.
84 = 2² × 3 × 7
180 = 2² × 3² × 5
HCF = 2² × 3
= 12
LCM = 2² × 3² × 5 × 7
= 1260
Verification:
12 × 1260 = 15120
84 × 180 = 15120
Verified.
Q2. Find the HCF and LCM of 300 and 150.
Ans.
300 = 2² × 3 × 5²
150 = 2 × 3 × 5²
HCF = 2 × 3 × 5²
= 150
LCM = 2² × 3 × 5²
= 300
Q3. Find the HCF and LCM of 630 and 770.
Ans.
630 = 2 × 3² × 5 × 7
770 = 2 × 5 × 7 × 11
HCF = 2 × 5 × 7
= 70
LCM = 2 × 3² × 5 × 7 × 11
= 6930
Most Important Questions from Class 7 Maths Chapter 11 for 2026 Exams
Use this section for last revision. These questions cover calculation, property, and application patterns.
Find the HCF of 96 and 275.
Ans. HCF = 1
Find the LCM of 105, 195, and 65.
Ans. 105 = 3 × 5 × 7, 195 = 3 × 5 × 13, 65 = 5 × 13
LCM = 3 × 5 × 7 × 13
= 1365
Find two numbers with HCF 1 and LCM 66.
Ans. 6 and 11
The HCF of two numbers is 12 and their LCM is 360. One number is 60. Find the other number.
Ans.
HCF × LCM = Product of numbers
12 × 360 = 60 × other number
Other number = 4320/60
= 72
Verify HCF × LCM = product for 45 and 105.
Ans.
45 = 3² × 5
105 = 3 × 5 × 7
HCF = 15
LCM = 315
15 × 315 = 4725
45 × 105 = 4725
Verified.
Children play “Fire in the Mountain.” When 6 is called, no one is out. When 9 is called, no one is out. When 10 is called, some are out. How many children could be playing?
Ans. The number must be divisible by 6 and 9 but not by 10.
LCM of 6 and 9 = 18
Possible numbers: 18, 36, 54, 72, 90
But 90 is divisible by 10.
So, possible answers include 18, 36, 54, or 72.
What is the LCM of two different prime numbers m and n?
Ans. The LCM is m × n because two different prime numbers are co-prime.
Important Formulas and Rules from Class 7 Maths Chapter 11
| Concept |
Rule |
| HCF |
Highest common factor |
| LCM |
Lowest common multiple |
| HCF by prime factorisation |
Common primes with minimum powers |
| LCM by prime factorisation |
All primes with maximum powers |
| Co-prime numbers |
HCF = 1 |
| Consecutive numbers |
HCF = 1 |
| One number factor of another |
HCF = smaller number |
| One number multiple of another |
LCM = larger number |
| Product property |
HCF × LCM = product of two numbers |