Class 7 Maths Chapter 3 Finding Common Ground worksheet with MCQs, short answer questions, practice questions and extra questions based on Ganita Prakash Part 2 for revision and exam preparation.

Class 7 Maths Chapter 3 Finding Common Ground Worksheet

Finding Common Ground

Class 7 Mathematics | Ganita Prakash Part II | Chapter 3

Worksheet: Factors, Multiples, HCF and LCM

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Section A: Factors and Multiples

1. Fill in the blanks.

(a) A number that divides another number exactly is called a __________.

(b) A number obtained by multiplying a number by a whole number is called a __________.

(d) The smallest prime number is __________.

(e) A prime number has exactly __________ factors.

2. Write all the factors.

(a) Factors of 24:

(b) Factors of 36:

(d) Factors of 60:

3. Write the first six multiples.

(a) Multiples of 7:

(b) Multiples of 9:

4. State whether each statement is True or False.

(a) 5 is a factor of 35. True / False

(b) 8 is a factor of 30. True / False

(d) 25 is a multiple of 4. True / False

(e) 2 is the only even prime number. True / False

Section B: Prime Numbers and Prime Factorisation

5. Circle the prime numbers.

2 4 5 9 11 15 17 21 23 29

6. Write the prime factorisation.

(a) 36 =

(b) 48 =

(d) 90 =

(e) 120 =

7. Use prime factorisation to find the factors of 100.

Prime factorisation:

All factors:

8. Answer the following.

(a) Is 14 a factor of 840? Explain.

(b) Is 27 a factor of 840? Explain.

Section C: HCF

9. Find the HCF by listing the factors.

(a) 18 and 24

(b) 30 and 45

10. Find the HCF using prime factorisation.

(a) 45 and 75

(b) 84 and 112

(d) 225 and 750

11. Find the common factors and HCF.

Numbers	Common Factors	HCF
50, 60
140, 275
81, 243
72, 108

12. Word Problem

A rectangular floor is 18 m long and 24 m wide. It is to be covered with square tiles of the same size. The side of each tile must be a whole number of metres. What is the largest possible side length of each square tile?

13. Word Problem

A shopkeeper has 84 kg of rice and 108 kg of wheat. He wants to pack both separately into bags of equal whole-number weight and wants to use the fewest bags possible. What should be the weight of each bag?

Section D: LCM

14. Find the LCM by listing multiples.

(a) 6 and 8

(b) 9 and 12

15. Find the LCM using prime factorisation.

(a) 14 and 35

(b) 36 and 54

(d) 96 and 360

16. Complete the table.

Numbers	HCF	LCM
12, 18
15, 25
20, 30
24, 36
18, 45

17. Word Problem

One bell rings every 6 minutes and another bell rings every 8 minutes. If both bells ring together at 9:00 a.m., after how many minutes will they ring together again?

18. Word Problem

A student visits the library every 10 days and attends a mathematics club every 15 days. If both activities happen today, after how many days will they happen on the same day again?

Section E: HCF and LCM Properties

19. Complete the statements.

(a) If one number is a factor of another, the HCF is __________.

(b) If one number is a factor of another, the LCM is __________.

(d) The LCM of two co-prime numbers is their __________.

(e) For two positive integers:

HCF × LCM = ______________________________

20. Find the missing number.

HCF of 18 and 24 = 6. If their LCM is 72, verify:

HCF × LCM = 18 × 24

Show your calculation.

21. Find the missing value.

(a) HCF = 5, LCM = 60, one number = 15. Find the other number.

(b) HCF = 7, LCM = 84, one number = 21. Find the other number.

22. State whether the following are True or False.

(a) HCF of two consecutive numbers is 1. True / False

(b) LCM of two different prime numbers is their product. True / False

(d) LCM can be greater than the product of two positive integers. True / False

(e) If one number divides another, the larger number is their LCM. True / False

Section F: Reasoning and Generalisation

23. Answer with reasoning.

Can two consecutive odd numbers have an HCF greater than 1? Explain with a reason.

24. Find a counterexample.

A student says:

“The larger a number is, the longer its prime factorisation must be.”

Give two numbers that show this statement is not always true.

25. Generalise.

If one number is n and the other number is kn, where k is a positive integer, write the HCF and LCM in algebraic form.

26. Think and answer.

Two numbers have HCF 1. Does this always mean that both numbers are prime? Explain with an example.

27. Challenge Question

Find two numbers whose HCF is 1 and LCM is 66. Write all possible pairs of positive integers.

Section G: Mixed Practice

28. Multiple Choice Questions

(a) Which is the HCF of 36 and 48?

(i) 6 (ii) 12 (iii) 18 (iv) 24

(b) Which is the LCM of 12 and 18?

(i) 24 (ii) 30 (iii) 36 (iv) 54

(c) Which number is a factor of 225?

(i) 14 (ii) 15 (iii) 16 (iv) 17

(d) The LCM of two different prime numbers m and n is:

(i) m+n (ii) m−n (iii) mn (iv) 1

(e) Which of the following is the prime factorisation of 180?

(i) 2 × 2 × 3 × 3 × 5

(ii) 2 × 3 × 5

(iii) 2 × 2 × 2 × 3 × 5

(iv) 3 × 3 × 5 × 5

29. Solve.

Find the smallest number that is divisible by 3, 4, 5 and 6.

30. Solve.

Find the largest number that divides 84 and 126 exactly.

31. Solve.

Find the smallest number that is a multiple of 8, 12 and 18.

32. Solve.

Find the HCF and LCM of 90 and 150. Then verify the relationship:

HCF × LCM = Product of the two numbers

33. Fractions and LCM

Find the LCM of 12, 18 and 30. Explain why finding the LCM is useful when adding fractions with these denominators.

Section H: Higher Order Thinking

34.

The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other number.

35.

A number is less than 100 and is divisible by 6, 8 and 9. Find the number.

36.

A school has 48 boys and 60 girls. They have to be divided into groups so that every group has the same number of boys and the same number of girls, with the largest possible number of groups. Find the number of groups and the number of boys and girls in each group.

37.

Two lights flash every 12 seconds and 18 seconds. They flash together at the start. After how many seconds will they flash together again?

38. Challenge

Find the smallest number greater than 100 that is divisible by 4, 6, 8 and 10.

Answer Key

1. (a) factor (b) multiple (c) prime, composite (d) 2 (e) 2

2. (a) 1, 2, 3, 4, 6, 8, 12, 24
(b) 1, 2, 3, 4, 6, 9, 12, 18, 36
(c) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
(d) 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

3. (a) 7, 14, 21, 28, 35, 42
(b) 9, 18, 27, 36, 45, 54
(c) 12, 24, 36, 48, 60, 72

4. (a) True
(b) False
(c) True
(d) False
(e) True

5. 2, 5, 11, 17, 23, 29

6. (a) 36 = 2 × 2 × 3 × 3
(b) 48 = 2 × 2 × 2 × 2 × 3
(c) 72 = 2 × 2 × 2 × 3 × 3
(d) 90 = 2 × 3 × 3 × 5
(e) 120 = 2 × 2 × 2 × 3 × 5

7. 100 = 2 × 2 × 5 × 5
Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100

8. (a) Yes, 840 ÷ 14 = 60.
(b) No, 840 is not divisible by 27.
(c) Yes, 840 ÷ 8 = 105.

9. (a) HCF(18,24) = 6
(b) HCF(30,45) = 15
(c) HCF(36,60) = 12

10. (a) 15
(b) 28
(c) 48
(d) 75

11. 50, 60 → Common factors: 1, 2, 5, 10 → HCF = 10
140, 275 → Common factors: 1, 5 → HCF = 5
81, 243 → Common factors: 1, 3, 9, 27, 81 → HCF = 81
72, 108 → Common factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 → HCF = 36

12. HCF(18,24) = 6.
Largest square tile = 6 m × 6 m.

13. HCF(84,108) = 12.
Each bag should contain 12 kg.

14. (a) 24
(b) 36
(c) 30

15. (a) 70
(b) 108
(c) 360
(d) 1440

16. 12, 18 → HCF = 6, LCM = 36
15, 25 → HCF = 5, LCM = 75
20, 30 → HCF = 10, LCM = 60
24, 36 → HCF = 12, LCM = 72
18, 45 → HCF = 9, LCM = 90

17. LCM(6,8) = 24 minutes.
They ring together again at 9:24 a.m.

18. LCM(10,15) = 30 days.

19. (a) smaller number
(b) larger number
(c) 1
(d) product
(e) product of the two numbers

20. 6 × 72 = 432 and 18 × 24 = 432.
Therefore the relationship is verified.

21. (a) Other number = 20
(b) Other number = 28

22. (a) True
(b) True
(c) False
(d) False
(e) True

23. No. Two consecutive odd numbers always have HCF 1 because any common factor of two consecutive numbers must divide their difference, which is 2. Since both numbers are odd, 2 cannot be their common factor.

24. For example, 96 = 2 × 2 × 2 × 2 × 2 × 3, while 121 = 11 × 11. Although 121 is larger, its prime factorisation has fewer prime factors.

25. HCF(n, kn) = n
LCM(n, kn) = kn

26. No. For example, 8 and 9 are both composite, but HCF(8,9) = 1.

27. The possible pairs are: (1, 66), (2, 33), (3, 22), (6, 11).

28. (a) (ii) 12
(b) (iii) 36
(c) (ii) 15
(d) (iii) mn
(e) (i) 2 × 2 × 3 × 3 × 5

29. LCM(3,4,5,6) = 60.

30. HCF(84,126) = 42.

31. LCM(8,12,18) = 72.

32. HCF(90,150) = 30
LCM(90,150) = 450
30 × 450 = 13,500
90 × 150 = 13,500

33. LCM(12,18,30) = 180.
The LCM gives a common denominator that can be used to rewrite the fractions before adding them.

34. HCF × LCM = Product
12 × 180 = 36 × x
2160 = 36x
x = 60

35. LCM(6,8,9) = 72.
The number less than 100 is 72.

36. HCF(48,60) = 12.
There can be 12 groups.
Each group has 4 boys and 5 girls.

37. LCM(12,18) = 36 seconds.

38. LCM(4,6,8,10) = 120.
The smallest number greater than 100 is therefore 120.

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Class 7 Maths Chapter 3 Worksheet | Finding Common Ground