Find the best tutors and institutes for Class 10 Tuition
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
Solution :
Given: Two integers, $a = 510$ and $b = 92$.
To Find: The Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the given integers, and to verify the relationship: $LCM \times HCF = \text{Product of the two numbers}$.
Step 1: Prime Factorization of the given numbers
We express each number as a product of its prime factors.
For $510$:
$510 = 2 \times 255$
$255 = 3 \times 85$
$85 = 5 \times 17$
$17 = 17 \times 1$
So, $510 = 2^1 \times 3^1 \times 5^1 \times 17^1$
For $92$:
$92 = 2 \times 46$
$46 = 2 \times 23$
$23 = 23 \times 1$
So, $92 = 2^2 \times 23^1$
Step 2: Determining HCF and LCM
HCF (Highest Common Factor): The HCF is the product of the smallest power of each common prime factor in the numbers.
Common prime factor is $2$. The smallest power of $2$ is $2^1$.
$HCF(510, 92) = 2^1 = 2$.
LCM (Least Common Multiple): The LCM is the product of the greatest power of each prime factor involved in the numbers.
Prime factors involved are $2, 3, 5, 17, 23$.
Greatest powers are $2^2, 3^1, 5^1, 17^1, 23^1$.
$LCM(510, 92) = 2^2 \times 3^1 \times 5^1 \times 17^1 \times 23^1$
$LCM(510, 92) = 4 \times 3 \times 5 \times 17 \times 23$
$LCM(510, 92) = 12 \times 5 \times 17 \times 23$
$LCM(510, 92) = 60 \times 17 \times 23$
$LCM(510, 92) = 1020 \times 23$
$LCM(510, 92) = 23460$.
Step 3: Verification of the relationship
We need to verify if $LCM \times HCF = a \times b$.
Left Hand Side (LHS):
$LCM \times HCF = 23460 \times 2$
$LCM \times HCF = 46920$.
Right Hand Side (RHS):
Product of numbers $= 510 \times 92$
$510 \times 92 = 510 \times (90 + 2)$
$= 45900 + 1020$
$= 46920$.
Since $LHS = RHS$, the relationship is verified.
Final Answer: The HCF is $2$, the LCM is $23460$, and the relationship $LCM \times HCF = \text{Product of the two numbers}$ is verified as $46920 = 46920$.
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(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
- 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
Top Tutors who teach Real numbers
Top Tutor
Patel Nagar, Gurgaon
13 yrs of
Exp
300
per hour
Online Classes
Student's Home
Tutor's home
Best science teacher. I never had interest in science , since when she started teaching me i improved a lot and finally got interest in science.
Top Tutor
Nerul, Mumbai
6 yrs of
Exp
1 student
355 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have specifically guided Class 10 students for board exams, with a strong focus on NCERT syllabus, and topics like Real Numbers, Polynomials, Quadratic Equations, and Statistics. I provide regular tests, personalized feedback, and exam-oriented strategies to improve accuracy and speed. Many of my students have shown remarkable improvement and scored above 90% in their board exams.
Sir is very nice in explaining the concepts . He gives many question for me to practice and explains them in a simple and straightforward manner.
Top Tutor
Doddakallasandra, Bangalore
11 yrs of
Exp
3 students
14 hrs of classes completed
500
per hour
Tutor's home
Have taught in school for 10th grade students.
Capability to handle problems & explain them to the last detail is very we’ll appreciated. Very detailed and informative on IGCSE maths - would definitely recommend him.
Top Tutor
Indirapuram, Ghaziabad
5 yrs of
Exp
39 hrs of classes completed
Online Classes
I was a working professional .Left my job and started teaching as a tutor. I have been a national level debater . I am a famous teacher amongst my students .
Top Tutor
Sector-21, Chandigarh
6 yrs of
Exp
150
per hour
Online Classes
Student's Home
Tutor's home
All the concepts are taught well and doubts are resolved at the earliest. Each student gets personal attention.for in depth knowledge and concept clarity , students of targeted classes may contact. M well experienced, qualified teacher and tutor with over ,6 years of experience in teaching maths and chemistry across different boards including cbse ,icse M a passionate teacher helping hundreds of students to overcome their fear from maths nd chemistry
I am very much thankful to Ravinder sir. My son scored very good score in class 10 under his guidance. Their way of teaching is very impressive and very impactful. Doubts of my child we're cleared on time that's why he was able to complete his syllabus and he was able to get 9.8 CGPA. Their way of teaching is different from other institution in Chandigarh. I would highly recommend all the parents to send their child to excellence academy for their bright future.
Find more Tutor for Real numbers in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
