What Are LCM and HCF?

 

What Are LCM and HCF?

HCF (Highest Common Factor)

• Definition: The largest number that divides two or more numbers exactly, without leaving a remainder. Also called GCD (Greatest Common Divisor)126.

• Example: HCF of 24 and 36 is 12, because 12 is the biggest number that divides both 24 and 36 completely1.

LCM (Least Common Multiple)

• Definition: The smallest number that is a multiple of two or more numbers126.

• Example: LCM of 8 and 16 is 16, because 16 is the smallest number that both 8 and 16 divide into exactly1.

---

Why Are LCM and HCF Important?

• LCM is useful for finding common time intervals, aligning schedules, or adding fractions with different denominators67.

• HCF helps in simplifying fractions, dividing things equally, or finding the largest possible group size6.

---

How to Find LCM and HCF

Methods

1. Listing Method

   • List out all multiples (for LCM) or all factors (for HCF) and find the common one.

2. Prime Factorization Method

   • Break each number into prime factors.

   • HCF: Multiply the lowest power of all common prime factors.

   • LCM: Multiply the highest power of all prime factors present in any number2357.

3. Division Method

   • For HCF: Divide the larger number by the smaller, then divide the divisor by the remainder, and repeat until remainder is 0. The last divisor is the HCF25.

   • For LCM: LCM = (Product of numbers) ÷ HCF3.

---

Examples

Example 1: HCF

Find HCF of 18 and 24.

• 18 = 2 × 3 × 3

• 24 = 2 × 2 × 2 × 3

• Common prime factors: 2 and 3

• HCF = 2 × 3 = 6

Example 2: LCM

Find LCM of 12 and 15.

• 12 = 2 × 2 × 3

• 15 = 3 × 5

• All primes: 2, 3, 5

• LCM = 2 × 2 × 3 × 5 = 60

Example 3: Relationship

If HCF of two numbers is 4 and their LCM is 60, what could the numbers be?

• Let numbers be 4 × a and 4 × b, where a and b are co-prime.

• LCM = 4 × a × b = 60 ⇒ a × b = 15 ⇒ possible pairs: (1,15), (3,5)

• Numbers: 4 & 60, or 12 & 20

---

Quick Reference Table

Aspect	LCM (Least Common Multiple)	HCF (Highest Common Factor)
Definition	Smallest common multiple	Largest common factor
Use	Aligning cycles, adding fractions	Simplifying, dividing equally
Example (4,6)	12	2
Methods	Listing, Prime Factorization, Division	Listing, Prime Factorization, Division

---

100 Practice Questions with Answers and Explanations

Questions 1-20: Find HCF

1. HCF of 12 and 18

2. HCF of 24 and 36

3. HCF of 15 and 25

4. HCF of 28 and 42

5. HCF of 35 and 49

6. HCF of 16 and 24

7. HCF of 21 and 28

8. HCF of 32 and 40

9. HCF of 14 and 35

10. HCF of 18 and 27

11. HCF of 45 and 60

12. HCF of 81 and 27

13. HCF of 100 and 80

14. HCF of 56 and 98

15. HCF of 33 and 44

16. HCF of 48 and 180

17. HCF of 50 and 75

18. HCF of 63 and 84

19. HCF of 72 and 120

20. HCF of 90 and 150

Questions 21-40: Find LCM

21. LCM of 3 and 5

22. LCM of 4 and 6

23. LCM of 8 and 12

24. LCM of 7 and 14

25. LCM of 9 and 12

26. LCM of 10 and 15

27. LCM of 5 and 20

28. LCM of 6 and 9

29. LCM of 12 and 18

30. LCM of 16 and 24

31. LCM of 14 and 21

32. LCM of 18 and 24

33. LCM of 20 and 30

34. LCM of 25 and 40

35. LCM of 28 and 35

36. LCM of 32 and 48

37. LCM of 36 and 54

38. LCM of 42 and 63

39. LCM of 45 and 60

40. LCM of 48 and 72

Questions 41-60: Mixed

41. HCF and LCM of 8 and 20

42. HCF and LCM of 9 and 27

43. HCF and LCM of 14 and 35

44. HCF and LCM of 18 and 24

45. HCF and LCM of 15 and 45

46. HCF and LCM of 21 and 49

47. HCF and LCM of 16 and 36

48. HCF and LCM of 24 and 60

49. HCF and LCM of 30 and 45

50. HCF and LCM of 12 and 20

51. HCF and LCM of 10 and 25

52. HCF and LCM of 22 and 33

53. HCF and LCM of 27 and 36

54. HCF and LCM of 32 and 56

55. HCF and LCM of 18 and 30

56. HCF and LCM of 35 and 50

57. HCF and LCM of 40 and 60

58. HCF and LCM of 28 and 42

59. HCF and LCM of 36 and 48

60. HCF and LCM of 20 and 32

Questions 61-80: Word Problems

61. Two ropes are 12 m and 18 m long. What is the greatest length that can be measured exactly using both ropes?

62. Find the least number which is exactly divisible by 6, 8, and 12.

63. Three bells ring at intervals of 4, 5, and 6 minutes. When will they ring together again?

64. Find the largest number that divides 40, 56, and 72 exactly.

65. What is the smallest number which leaves a remainder 1 when divided by 2, 3, and 4?

66. Find the greatest number that will divide 148, 246, and 590 leaving the same remainder.

67. Find the least number which is a multiple of 9, 12, and 15.

68. Find the HCF of 0 and 15.

69. Find the LCM of 0 and 20.

70. Find the smallest number divisible by 5, 6, 8, and 12.

71. Find the largest number that divides 125, 175, and 225 exactly.

72. Two numbers have HCF 13 and LCM 455. If one number is 65, find the other.

73. Find the least number which when divided by 8, 12, and 16 leaves a remainder 5 in each case.

74. The product of two numbers is 180. If their HCF is 6, find their LCM.

75. Find the greatest number that divides 30 and 78 leaving remainder 6 in each case.

76. Find the least number which is exactly divisible by all numbers from 1 to 10.

77. Find the HCF of 2.5 and 0.5.

78. Find the LCM of 0.25 and 0.5.

79. Find the least number which when divided by 7, 8, and 9 leaves a remainder 2 in each case.

80. Find the largest number that divides 245 and 367 leaving remainder 5 in each case.

Questions 81-100: Challenging

81. Find the HCF and LCM of 72, 108, and 180.

82. Find the LCM of first five odd numbers.

83. Find the HCF of first four even numbers.

84. If the HCF and LCM of two numbers are 8 and 96 respectively, and one number is 32, find the other.

85. Find the LCM of 2, 3, 4, 5, 6.

86. Find the HCF of 77, 121, and 143.

87. Find the LCM of 15, 20, and 24.

88. If the product of two numbers is 540 and their HCF is 6, find their LCM.

89. Find the HCF and LCM of 0.6 and 1.8.

90. Find the smallest number divisible by 12, 15, 20, and 27.

91. Find the greatest number that divides 148, 246, and 590 leaving the same remainder.

92. Find the least number which when divided by 8, 12, and 16 leaves a remainder 5 in each case.

93. Find the HCF and LCM of 84 and 120.

94. Find the LCM of 3, 7, and 14.

95. Find the HCF of 45, 60, and 75.

96. Find the LCM of 2.5, 0.5, and 1.

97. Find the HCF of 0.3, 1.2, and 2.1.

98. Find the least number which when divided by 6, 7, 8, 9, and 12 leaves 1 in each case.

99. Find the HCF and LCM of 125 and 625.

100. Find the LCM of 18, 24, 36, and 45.

---

Answers and Explanations

1-20 HCF Answers

1. 6 (Common factors: 1,2,3,6)

2. 12 (Common factors: 1,2,3,4,6,12)

3. 5 (Common factors: 1,5)

4. 14 (Common factors: 1,2,7,14)

5. 7 (Common factors: 1,7)

6. 8 (Common factors: 1,2,4,8)

7. 7 (Common factors: 1,7)

8. 8 (Common factors: 1,2,4,8)

9. 7 (Common factors: 1,7)

10. 9 (Common factors: 1,3,9)

11. 15 (Common factors: 1,3,5,15)

12. 27 (Common factors: 1,3,9,27)

13. 20 (Common factors: 1,2,4,5,10,20)

14. 14 (Common factors: 1,2,7,14)

15. 11 (Common factors: 1,11)

16. 12 (Common factors: 1,2,3,6,12)

17. 25 (Common factors: 1,5,25)

18. 21 (Common factors: 1,3,7,21)

19. 24 (Common factors: 1,2,3,4,6,8,12,24)

20. 30 (Common factors: 1,2,3,5,6,10,15,30)

21-40 LCM Answers

21. 15 (Smallest number divisible by 3 and 5)

22. 12 (Smallest number divisible by 4 and 6)

23. 24 (Smallest number divisible by 8 and 12)

24. 14 (Smallest number divisible by 7 and 14)

25. 36 (Smallest number divisible by 9 and 12)

26. 30 (Smallest number divisible by 10 and 15)

27. 20 (Smallest number divisible by 5 and 20)

28. 18 (Smallest number divisible by 6 and 9)

29. 36 (Smallest number divisible by 12 and 18)

30. 48 (Smallest number divisible by 16 and 24)

31. 42 (Smallest number divisible by 14 and 21)

32. 72 (Smallest number divisible by 18 and 24)

33. 60 (Smallest number divisible by 20 and 30)

34. 200 (Smallest number divisible by 25 and 40)

35. 140 (Smallest number divisible by 28 and 35)

36. 96 (Smallest number divisible by 32 and 48)

37. 108 (Smallest number divisible by 36 and 54)

38. 126 (Smallest number divisible by 42 and 63)

39. 180 (Smallest number divisible by 45 and 60)

40. 144 (Smallest number divisible by 48 and 72)

41-60 Mixed Answers

41. HCF: 4, LCM: 40

42. HCF: 9, LCM: 27

43. HCF: 7, LCM: 70

44. HCF: 6, LCM: 72

45. HCF: 15, LCM: 45

46. HCF: 7, LCM: 147

47. HCF: 4, LCM: 144

48. HCF: 12, LCM: 120

49. HCF: 15, LCM: 90

50. HCF: 4, LCM: 60

51. HCF: 5, LCM: 50

52. HCF: 11, LCM: 66

53. HCF: 9, LCM: 108

54. HCF: 8, LCM: 224

55. HCF: 6, LCM: 90

56. HCF: 5, LCM: 350

57. HCF: 20, LCM: 120

58. HCF: 14, LCM: 84

59. HCF: 12, LCM: 144

60. HCF: 4, LCM: 160

61-80 Word Problems Answers

61. 6 m (HCF of 12 and 18)

62. 24 (LCM of 6, 8, 12)

63. 60 min (LCM of 4, 5, 6)

64. 8 (HCF of 40, 56, 72)

65. 13 (LCM of 2, 3, 4 is 12; 12+1=13)

66. 49 (HCF of differences: 246-148=98, 590-246=344, HCF of 98 and 344 is 49)

67. 180 (LCM of 9, 12, 15)

68. 15 (Any number with 0 and another, HCF is the non-zero number)

69. 0 (LCM with 0 is always 0)

70. 120 (LCM of 5, 6, 8, 12)

71. 25 (HCF of 125, 175, 225)

72. 91 (LCM × HCF = product of numbers; 65 × other = 13 × 455 → other = 91)

73. 173 (LCM of 8, 12, 16 = 48; 48+5=53, next is 101, next is 149, next is 197, so 173 is the answer)

74. 30 (Product = HCF × LCM; 180 = 6 × LCM ⇒ LCM = 30)

75. 12 (Subtract 6 from each: 24 and 72; HCF of 24 and 72 is 12)

76. 2520 (LCM of 1 to 10)

77. 0.5 (Convert to fractions: 2.5, 0.5 → 5/2, 1/2; HCF is 1/2)

78. 0.5 (LCM of 0.25, 0.5)

79. 146 (LCM of 7, 8, 9 = 504; 504+2=506, next is 1010, next is 1514, so 146 is the answer)

80. 122 (245-5=240, 367-5=362; HCF of 240 and 362 is 122)

81-100 Challenging Answers

81. HCF: 12, LCM: 1080

82. 315 (LCM of 1, 3, 5, 7, 9)

83. 2 (HCF of 2, 4, 6, 8)

84. 24 (LCM × HCF = product of numbers; 32 × other = 8 × 96 → other = 12)

85. 60 (LCM of 2, 3, 4, 5, 6)

86. 11 (HCF of 77, 121, 143)

87. 120 (LCM of 15, 20, 24)

88. 90 (Product = HCF × LCM; 540 = 6 × LCM → LCM = 90)

89. HCF: 0.6, LCM: 1.8

90. 540 (LCM of 12, 15, 20, 27)

91. 49 (Same as Q66)

92. 173 (Same as Q73)

93. HCF: 12, LCM: 420

94. 42 (LCM of 3, 7, 14)

95. 15 (HCF of 45, 60, 75)

96. 2.5 (LCM of 2.5, 0.5, 1)

97. 0.3 (HCF of 0.3, 1.2, 2.1)

98. 5041 (LCM of 6, 7, 8, 9, 12 = 504; 504+1=505, next is 1009, next is 1513, next is 2017, next is 2521, next is 3025, next is 3529, next is 4033, next is 4537, next is 5041)

99. HCF: 125, LCM: 625

100. 360 (LCM of 18, 24, 36, 45)

---

Explanations for Difficult Problems

Q73:
LCM of 8, 12, 16 is 48. Any number of the form 48k+5 will leave remainder 5 when divided by 8, 12, and 16. The smallest such number >16 is 53, next is 101, etc. The answer is 53.

Q76:
LCM of 1 to 10 is 2520. This is the smallest number divisible by all numbers from 1 to 10.

Q98:
LCM of 6, 7, 8, 9, 12 is 504. So, the required number is 504 + 1 = 505, next is 1009, etc., until you find a number that fits all conditions. 5041 is the answer.

---

Tip:

• For LCM, always take the highest power of each prime.

• For HCF, always take the lowest power of each common prime.

• Use prime factorization for clarity and accuracy.

If you need more explanations or have specific questions, ask anytime!