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Q4:
Given that HCF (306, 657) = 9, find LCM (306, 657).
Solution :
Given:
Two positive integers, $a = 306$ and $b = 657$.
The Highest Common Factor (HCF) of the two integers is given as: $HCF(306, 657) = 9$.
To Find:
The Least Common Multiple (LCM) of the two integers, $LCM(306, 657)$.
Step 1: Stating the Fundamental Theorem/Relationship
For any two positive integers $a$ and $b$, the relationship between their HCF and LCM is defined by the following mathematical identity:
$HCF(a, b) \times LCM(a, b) = a \times b$
[This property holds true for any two positive integers.]
Step 2: Substituting the Known Values
Substitute the given values into the identity:
$a = 306$
$b = 657$
$HCF(306, 657) = 9$
$9 \times LCM(306, 657) = 306 \times 657$
Step 3: Isolating the LCM
To find the $LCM(306, 657)$, divide both sides of the equation by $9$:
$LCM(306, 657) = \frac{306 \times 657}{9}$
Step 4: Performing the Arithmetic Calculation
First, simplify the fraction by dividing $306$ by $9$:
$306 \div 9 = 34$
[Since $9 \times 30 = 270$ and $306 - 270 = 36$, and $9 \times 4 = 36$, therefore $30 + 4 = 34$]
Now, multiply the result by $657$:
$LCM(306, 657) = 34 \times 657$
Step 5: Final Multiplication
$34 \times 657 = 34 \times (600 + 50 + 7)$
$= (34 \times 600) + (34 \times 50) + (34 \times 7)$
$= 20400 + 1700 + 238$
$= 22100 + 238$
$= 22338$
Final Answer: The LCM(306, 657) is 22338.
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(i): Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21
- 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
- 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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