Here, students will find HCF and LCM questions with in-depth explanations that will aid in their understanding of the concept. In mathematics, LCM represents the Least Common Multiple and HCF represents Highest Common Factor. The HCF and LCM questions that we have provided here will assist students comprehend how to find the LCM and HCF of the given numbers. Different questions are also provided so that students can practise them, and they can compare their answers with the provided solutions on our page. For additional information on the HCF and LCM, click here.
HCF and LCM Questions with Solutions
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HCF Definition: The Highest Common Factor (HCF) of 2 or more numbers is defined as the greatest possible number that divides both the numbers completely. For example, if “a” and “b” are the two numbers, then the HCF of a and b, i.e, HCF (a, b) is the greatest possible number that divides both a and b. The two most commonly used methods to find the HCF of the given numbers are:
Also, read: HCF. |
1. Find the HCF of 24 and 36.
Solution:
To find HCF of 24 and 36, first list the factors of 24 and 36.
Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Here, the greatest factor that is common in both the factors list is 12.
Thus, 12 is the greatest number that divides both 24 and 36 evenly.
Therefore, the HCF of 24 and 36 is 12.
2. Determine the GCF of 18 and 48.
Solution:
To find: HCF (18, 48)
The factors of 18 are 1, 2, 3, 6, 9 and 18
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Thus, 6 is the greatest number that divides 18 and 48 completely.
Hence, the GCF/HCF of 18 and 48 is 6.
3. Find the HCF of 135 and 225 using the prime factorisation method.
Solution:
To find the HCF of 135 and 225 using the prime factorisation method, follow the below steps:
Thus, the prime factorisation of 135 is 3 × 3 × 3 × 5
The prime factorisation of 225 is 3 × 3 × 5 × 5.
Hence, the prime factors that is common in both the numbers is 3 × 3 × 5
Therefore, the product of prime factors of 3 × 3 × 5 is 45.
Hence, the HCF of 135 and 225 is 45.
4. What is the HCF of 408 and 1032?
Solution:
To find: HCF (408, 1032).
The prime factorisation of 408 is 2 × 2 × 2 × 3 × 17
The prime factorisation of 1032 is 2 × 2 × 2 × 3 × 43
Thus, the product of the prime factors of 2 × 2 × 2 × 3 is 24
Hence, the HCF of 408 and 1032 is 24.
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LCM Definition: The Least Common Multiple (LCM) of 2 or more given numbers is defined as the smallest common multiple, which is found in both the numbers. The LCM of any given number can also be found using the listing multiples method and prime factorisation method. Also, read: LCM. |
5. Find the LCM of 3 and 4.
Solution:
To find the LCM of 3 and 4 using the listing method, first list out the multiples of 3 and 4.
Thus, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, …
From the above list, 12 is the smallest number that is common in both the multiples list.
Therefore, the LCM of 3 and 4 is 12.
6. What is the LCM of 6 and 8?
Solution:
To find: LCM (6, 8).
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, …
The multiples of 8 are 8, 16, 24, 32, 40, 48, ….
Thus, the smallest common multiple of 6 and 8 is 24.
Therefore, the LCM of 6 and 8 is 24.
7. Determine the LCM of 4 and 12 using the prime factorisation method.
Solution:
To find the LCM of 4 and 12 using the prime factorisation method, follow the below steps.
Step 1: Find the prime factorization of given numbers:
The prime factorisation of 4 is 2 × 2
The prime factorisation of 12 is 2 × 2 × 3.
Step 2: The LCM of given numbers is found by multiplying the product of all factors. (Note: The common factor is included only once)
Hence, the product of prime factors = 2 × 2 × 3 = 12.
Therefore, the LCM of 4 and 12 is 12.
8. What is the LCM of 54 and 60?
Solution:
The prime factorisation of 54 is 2 × 3 × 3 × 3.
The prime factorisation of 60 is 2 × 2 × 3 × 5.
Thus, the product of prime factors = 2 × 2 × 3 × 3 × 3 × 5 = 540
Hence, the LCM of 54 and 60 is 540.
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Relation Between HCF and LCM: The product of given numbers is equal to the product of the HCF and LCM of natural numbers. I.e., LCM of given numbers × HCF of given numbers = Product of given numbers. To know more about the relation between HCF and LCM, click here. |
9. Prove that LCM(12, 7) × HCF(12, 7) = Product(12, 7).
Solution:
Given: LCM(12, 7) × HCF(12, 7) = Product(12, 7) …(1)
Finding LCM (12, 7):
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, ..
The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 120, …
Hence, the LCM of 12 and 7 is 84.
Finding HCF (12, 7):
The factors of 7 are 1 and 7.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
Hence, HCF of 12 and 7 is 1.
Finding Product of 12 and 7:
The product of 12 and 7 = 12 × 7 = 84.
Now, substitute the obtained values in (1), we get
84 × 1 = 84
84 = 84
Hence, LHS = RHS
Therefore, LCM(12, 7) × HCF(12, 7) = Product(12, 7) is proved.
10. Find the LCM of the fractions 2/5, 4/7 and 6/11.
Solution:
Given fractions: 2/5, 4/7 and 6/11
As we know, the formula to find the LCM of fractions is:
LCM of fractions = LCM of Numerators/HCF of Denominators .. (1)
Thus, the LCM of Numerators = LCM (2, 4, 6) = 12.
HCF of denominators = HCF (5, 7, 11) = 1
Now, substitute the values in (1), we get
LCM of fractions = 12/1 = 12
Hence, the LCM of the fractions 2/5, 4/7 and 6/11 is 12.
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Practice Questions
Solve the following HCF and LCM Questions:
- Find the LCM of 24 and 51.
- What is the GCF of 68 and 101?
- Find the LCM of fractions 11/6, 9/4, and 21/8.
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