Understanding HCF and LCM questions is a fundamental part of number theory in mathematics. These concepts help solve a variety of problems involving divisibility, time intervals, arrangement, and more.
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The HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly.
The LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.
HCF and LCM formulas, and a variety of solved HCF and LCM problems-including LCM and HCF for fractions and relationship-based problems, will all be covered below.
To solve HCF and LCM questions, it is important to use the correct formulas.
HCF (a, b) = Product of all common prime factors
HCF using division:
Continue dividing until the remainder is 0. The last divisor is the HCF.
LCM (a, b) = Product of all prime factors with the highest power
LCM (a, b) = (a × b) / HCF (a, b)
These HCF and LCM formulas are applicable to all types of HCF and LCM questions.
Find the HCF of 36 and 60 using prime factorization.
36 = 2² × 3²
60 = 2² × 3 × 5
Common prime factors: 2² × 3 = 12
HCF = 12
Find the HCF of 72 and 120 using division.
120 ÷ 72 = 1, remainder = 48
72 ÷ 48 = 1, remainder = 24
48 ÷ 24 = 2, remainder = 0
HCF = 24
Find the HCF of 45, 75, and 105.
45 = 3² × 5
75 = 3 × 5²
105 = 3 × 5 × 7
Common prime factors: 3 × 5 = 15
HCF = 15
What is the HCF of 16 and 28?
16 = 2⁴
28 = 2² × 7
Common prime factor: 2² = 4
HCF = 4
Find the LCM of 12 and 15.
12 = 2² × 3
15 = 3 × 5
LCM = 2² × 3 × 5 = 60
Find the LCM of 24 and 36.
24 = 2³ × 3
36 = 2² × 3²
LCM = 2³ × 3² = 72
Find the LCM of 14, 28, and 70.
14 = 2 × 7
28 = 2² × 7
70 = 2 × 5 × 7
LCM = 2² × 5 × 7 = 140
Find the LCM of 8 and 20 using the formula.
HCF of 8 and 20 = 4
LCM = (8 × 20) / 4 = 160 / 4 = 40
Problem:
Two pieces of wood are 96 cm and 144 cm long. What is the greatest length that can be cut from both pieces?
Solution:
We are looking for the greatest length that can divide both 96 and 144 exactly → Use HCF.
Prime factorization:
96 = 2⁵ × 3
144 = 2⁴ × 3²
Common factors = 2⁴ × 3 = 16 × 3 = 48
Answer: 48 cm
Problem:
Two alarm clocks ring every 15 minutes and 20 minutes. If both ring together at 12 PM, when will they ring together next?
Solution:
This is a case of events repeating together → Use LCM.
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes
Next ring together at: 12 PM + 60 min = 1:00 PM
Answer: In 60 minutes or at 1:00 PM
Problem:
A gardener has 90 rose plants and 150 jasmine plants. He wants to arrange them in equal rows with no plant left over. Find the maximum number of rows possible.
Solution:
Maximum number of rows → Use HCF.
Prime factorization:
90 = 2 × 3² × 5
150 = 2 × 3 × 5²
Common prime factors = 2 × 3 × 5 = 30
Answer: 30 rows
Problem:
Find the smallest number that is divisible by 8, 9, and 12.
Solution:
Smallest number divisible by all → Use LCM.
8 = 2³
9 = 3²
12 = 2² × 3
LCM = 2³ × 3² = 8 × 9 = 72
Answer: 72
To solve LCM and HCF for fractions, use these formulas:
HCF of fractions = HCF of numerators / LCM of denominators
LCM of fractions = LCM of numerators / HCF of denominators
Find the HCF and LCM of 2/3 and 4/5.
HCF of 2 and 4 = 2
LCM of 3 and 5 = 15
HCF = 2 / 15
LCM of 2 and 4 = 4
HCF of 3 and 5 = 1
LCM = 4 / 1 = 4
Find the HCF and LCM of 3/7 and 6/14.
HCF(3,6) = 3, LCM(7,14) = 14
HCF = 3 / 14
LCM(3,6) = 6, HCF(7,14) = 7
LCM = 6 / 7
The product of two numbers = HCF × LCM
This relation is useful in many HCF and LCM questions.
Question 1: If the HCF of two numbers is 12 and their LCM is 180, find the product of the numbers.
Solution:
We use the formula:
HCF × LCM = Product of the two numbers
= 12 × 180 = 2160
Answer: Product of the two numbers = 2160
Question 2: If the product of two numbers is 960 and their HCF is 12, find their LCM.
Solution:
HCF × LCM = Product of numbers
12 × LCM = 960
LCM = 960 ÷ 12 = 80
Answer: LCM = 80
Question 3: The LCM of two numbers is 120 and the HCF is 6. If one number is 30, find the other number.
Solution:
Use the relation:
Product of numbers = HCF × LCM = 6 × 120 = 720
Other number = 720 ÷ 30 = 24
Answer: The other number is 24
Question 4: Two numbers have HCF = 4 and LCM = 240. If one number is 60, find the other.
Solution:
Product of numbers = HCF × LCM = 4 × 240 = 960
Other number = 960 ÷ 60 = 16
Answer: The other number is 16
Question 5: The two numbers are 36 and 60. Verify that HCF × LCM = Product of the numbers.
Solution:
HCF(36, 60) = 12
LCM(36, 60) = 180
HCF × LCM = 12 × 180 = 2160
36 × 60 = 2160
Verified.
Answer: HCF × LCM = Product of numbers = 2160
Question 6: The HCF and LCM of two numbers are 11 and 770 respectively. If one number is 55, find the other.
Solution:
Product = HCF × LCM = 11 × 770 = 8470
Other number = 8470 ÷ 55 = 154
Answer: The other number is 154
Find the HCF and LCM of 45 and 60.
Find the LCM of 5, 10, and 15.
Find the HCF of 28 and 42.
Find the smallest number divisible by 6, 8, and 12.
Find the HCF and LCM of 3/4 and 9/10.
Two buses leave a station at the same time. One returns every 24 minutes, the other every 36 minutes. When will they meet again?
LCM = 72 minutes
A rectangular tile of length 18 cm and width 24 cm is to be used to cover a floor. Find the largest size of square tile that can be used.
HCF = 6 cm
You can solve mathematical problems involving divisibility, arrangement, and multiples more effectively if you comprehend and practice hcf and lcm questions. Verify all HCF and LCM formulas, solve HCF problems, try LCM problems, and have a clear understanding of how HCF and LCM questions relate to one another.
Learn the formulas for LCM and HCF for fractions, and practice a wide range of HCF and LCM examples on a regular basis.
HCF and LCM - Quickly grasp the basics with simple explanations and examples
HCF (Highest Common Factor) - Understand with step-by-step explanations
Integers Questions - Practice and strengthen your basics with solved problems
To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of two numbers:
HCF Method:
List the factors of both numbers and find the greatest one common to both
Or use the Euclidean algorithm (divide and take remainders)
LCM Method:
List multiples of both numbers and find the smallest one common to both
Or use the formula:
LCM(a, b) = (a × b) / HCF(a, b)
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
HCF = 12
LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers
HCF (Highest Common Factor): The greatest number that divides two or more numbers exactly
Using the Euclidean algorithm:
645 ÷ 473 = 1 remainder 172
473 ÷ 172 = 2 remainder 129
172 ÷ 129 = 1 remainder 43
129 ÷ 43 = 3 remainder 0
HCF = 43
Keep practicing HCF and LCM problems to master the concept. Explore more math lessons and solved examples with Orchids The International School to strengthen your foundation!