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Q3(i):
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21
Solution :
Given: The set of integers $\{12, 15, 21\}$.
To Find: The Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the given integers using the prime factorisation method.
Step 1: Prime Factorisation of the given integers
We express each integer as a product of its prime factors:
For $12$:
$12 = 2 \times 6$
$12 = 2 \times 2 \times 3$
$12 = 2^2 \times 3^1$
For $15$:
$15 = 3 \times 5$
$15 = 3^1 \times 5^1$
For $21$:
$21 = 3 \times 7$
$21 = 3^1 \times 7^1$
Step 2: Determining the HCF
The Highest Common Factor (HCF) is defined as the product of the smallest power of each common prime factor in the numbers.
Looking at the prime factors:
Common prime factor: $3$
Smallest power of $3$ present in all factorisations is $3^1$.
Therefore, $\text{HCF}(12, 15, 21) = 3^1 = 3$.
Step 3: Determining the LCM
The Least Common Multiple (LCM) is defined as the product of the greatest power of each prime factor involved in the numbers.
The prime factors involved are $2, 3, 5,$ and $7$.
Greatest power of $2$ is $2^2$.
Greatest power of $3$ is $3^1$.
Greatest power of $5$ is $5^1$.
Greatest power of $7$ is $7^1$.
Calculating the LCM:
$\text{LCM}(12, 15, 21) = 2^2 \times 3^1 \times 5^1 \times 7^1$
$\text{LCM}(12, 15, 21) = 4 \times 3 \times 5 \times 7$
$\text{LCM}(12, 15, 21) = 12 \times 5 \times 7$
$\text{LCM}(12, 15, 21) = 60 \times 7$
$\text{LCM}(12, 15, 21) = 420$
Final Answer: The HCF of 12, 15, and 21 is 3, and the LCM is 420.
More Questions from Class 10 Mathematics Real numbers EXERCISE 1.1
- Q1(i): Express each number as a product of its prime factors: (i) 140
- Q1(ii): Express each number as a product of its prime factors: (ii) 156
- Q1(iii): Express each number as a product of its prime factors: (iii) 3825
- Q1(iv): Express each number as a product of its prime factors: (iv) 5005
- Q1(v): Express each number as a product of its prime factors: (v) 7429
- Q2(i): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (i) 26 and 91
- Q2(ii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (ii) 510 and 92
- Q2(iii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (iii) 336 and 54
- Q3(ii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29
- Q3(iii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (iii) 8, 9 and 25
- Q4: Given that HCF (306, 657) = 9, find LCM (306, 657).
- Q5: Check whether $6^n$ can end with the digit 0 for any natural number $n$.
- Q6: Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
- Q7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
CBSE Solutions for Class 10 Mathematics Real numbers
Chapters in CBSE - Class 10 Mathematics
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