NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2 - Real Numbers
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  • NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2 - Real Numbers

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2 - Real Numbers

Updated on 30 May 2025, 02:16 PM IST

The foundation of every mathematical concept depends on mastering numerical principles and their attributes. The fundamental principle in mathematics demonstrates how to break numbers into prime number factors to understand their internal makeup for better calculation purposes. Prime factorisation serves as a valuable method that reveals how to find both common divisors and multiples. The technique establishes a solid foundation to recognize numerical connections between numbers and functions effectively with extensive calculations and real-world applications about time duration and quantitative and repetitive scenarios.

This Story also Contains

  1. NCERT Solutions Class 10 Maths Chapter 1: Exercise 1.2 PDF
  2. Access Solutions of Real Numbers Class 10 Chapter 1 Exercise: 1.2
  3. Topics covered in Chapter 1, Real Numbers: Exercise 1.2
  4. NCERT Solutions of Class 10 Subject Wise
  5. NCERT Exemplar Solutions of Class 10 Subject Wise
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2 - Real Numbers
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2 - Real Numbers

The exercise provided through NCERT Solutions for Class 10 Maths in the standard NCERT Books teaches students about prime factorization in combination with the Highest Common Factor and the Lowest Common Multiple. All concepts covered in this exercise become vital building blocks for both arithmetic and algebra. The instructional materials teach students two methods for solving complex problems and confirming procedural validity. The questions that use digits with factorials alongside divisibility tests help students to develop their logical thinking abilities and strengthen their grasp of real numbers and their characteristics.

NCERT Solutions Class 10 Maths Chapter 1: Exercise 1.2 PDF

Access Solutions of Real Numbers Class 10 Chapter 1 Exercise: 1.2

Q1 (1) Express each number as a product of its prime factors: 140

Answer:

By taking LCM of the number 140 we can express it as a product of its prime factors, like as follows:

$140=2\times 2\times 5\times 7$

$140=2^{2}\times 5\times 7$

Q 1 (2) Express each number as a product of its prime factors: 156

Answer:

By taking LCM of the number 156 we can express it as a product of its prime factors, like as follows:

$156=2\times 2\times 3\times 13$

$156=2^{2}\times 3\times 13$

Q1 (3) Express each number as a product of its prime factors: 3825

Answer:

By taking LCM of the number 3825 we can express it as a product of its prime factors, like as follows:

$3825=3\times 3\times 5\times 5\times 17$

$3825=3^{2}\times 5^{2}\times 17$

Q1 (4) Express each number as a product of its prime factors: 5005

Answer:

By taking LCM of the number 5005 we can express it as a product of its prime factors, like as follows:

$5005=5\times 7\times 11\times 13$

Q1 (5) Express each number as a product of its prime factors: 7429

Answer:

By taking LCM of the number 7429 we can express it as a product of its prime factors, like as follows:

$7429=17\times 19\times 23$

Q2 (1) Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers: 26 and 91

Answer:

Express the numbers as the product of their prime numbers.

26 = 2 × 13

91 = 7 × 13

HCF(26, 91) = 13

LCM(26, 91) = 2 × 7 × 13 = 182

Verification; HCF × LCM = 13 × 182 = 2366

26 × 91 = 2366

26 × 91 = HCF × LCM

Hence, verified.

Q2 (2) Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. 510 and 92

Answer:

The number can be expressed as the product of prime factors as

510 = 2 × 3 x 5 × 17

92 = 22 × 23

HCF (510, 92) = 2

LCM (510, 92) = 22 × 3 x 5 × 17 × 23 = 23460

Verification; HCF x LCM = 2 × 23460 = 46920

510 × 92 = 46920

510 × 92 = HCF × LCM

Hence, verified

Q2 (3) Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. 336 and 54

Answer:

Express the numbers as the product of their prime numbers.

336 = 24 × 3 × 7

54 = 2 × 33

HCF(336,54) = 2 x 3 = 6

LCM(336,54) = 24 × 33 × 7 = 3024

Verification; HCF × LCM = 6 × 3024 = 18144

336 × 54 = 18144

336 × 54 = HCF × LCM

Hence, verified

Q3 (1) Find the LCM and HCF of the following integers by applying the prime factorization method. 12, 15 and 21

Answer:

The numbers can be written as the product of their prime factors as follows

12 = 22 x 3

15 = 3 x 5

21 = 3 x 7

HCF = 3

LCM = 22 x 3 x 5 x 7 = 420

Q3 (2) Find the LCM and HCF of the following integers by applying the prime factorisation method. 17, 23 and 29

Answer:

The given numbers are written as the product of their prime factors as follows

17 = 1 × 17

23 = 1 × 23

29 = 1 × 29

HCF = 1

LCM = 17 × 23 × 29 = 11339

Q3 (3) Find the LCM and HCF of the following integers by applying the prime factorization method. 8, 9 and 25

Answer:

The given numbers are written as the product of their prime factors as follows

8 = 23

9 = 32

25 = 52

HCF = 1

LCM = 23 × 32 × 52 = 1800

Q4 Given that HCF (306, 657) = 9, find LCM (306, 657).

Answer:

We know: HCF × LCM = a × b

So, HCF (306, 657) × LCM (306, 657) = 306 × 657

$LCM (306, 657) = \frac{306\times 657}{HCF (306, 657) }$

$ LCM (306, 657) = \frac{306\times 657}{9}$

$LCM (306, 657) =22338$

Q5 Check whether $6 ^n$ can end with the digit 0 for any natural number n.

Answer:

By prime factorisation, we have

6n = 2n × 3n

If a number's prime factorisation includes at least 1 as the power of both 2 and 5, then the number will conclude with 0. We can infer that, for every value of n, 6n will terminate with the number 0 since the prime factorisation of 6n shows that the power of 5 equals 0.

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.

Answer:

7 × 11 × 13 + 13

= (7 × 11 + 1) × 13

= 78 × 13 or 2 × 3 × 132

Similarly, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5

= (7 × 6 × 4 × 3 × 2 × 1 + 1) × 5

= 5 × 1008

The number rule states that we can take at least two common numbers out of two, and after solving, we saw that both numbers were even, as a result of which the number is composite.

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?

Answer:

The time after which they meet again at the starting point will be equal to the LCM of the times they individually take to complete one round.

Time taken by Sonia = 18 = 2 × 32

Time taken by Ravi = 12 = 22 × 3

LCM (18, 12) = 22 × 32 = 36

Therefore, they would again meet at the starting point after 36 minutes.

Topics covered in Chapter 1, Real Numbers: Exercise 1.2

1. Prime Factorization: Individuals learn the process to decompose whole numbers into one-of-a-kind factors consisting of prime numbers. Students need prime factorization as the base for solving multiple mathematical problems that include fraction simplification and LCM and HCF problem-solving and number property analysis in arithmetic and algebra.

2. Highest Common Factor (HCF) and Least Common Multiple (LCM): The teaching portion shows students how to find the greatest common factor which divides multiple numbers as well as the smallest common multiple that works for multiple numbers. The tools serve as basic requirements for tackling problems linked to periodicity, arrangement and comparison.

3. Relationship between HCF and LCM: An important identity is introduced: HCF × LCM = Product of the two numbers. The identity HCF × LCM = Product of the two numbers serves as an accuracy verifier and increases understanding of numerical relationships.

4. Identification of Composite Numbers: Using factorisation, students verify composite numbers in given expressions while developing their ability to think analytically and logically.

5. Concept of End Digits in Powers: Students investigate how numbers behave in powers and discover the circumstances in which power expressions can reach particular digits, such as zeros. Number patterns together with modular arithmetic form an important connection with this topic.

6. Real-Life Application of LCM (Cyclic Events): The concept of LCM applies to everyday scheduling through time and repetition when two people encounter each other after walking on a round track.

Also Read:

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Also see:

CBSE Class 10th Syllabus 2025-26
Students can access the subject-wise CBSE Class 10 syllabus for the 2025–26 academic session.
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NCERT Solutions of Class 10 Subject Wise

Students must check the NCERT solutions for class 10 of the Mathematics and Science Subjects.

NCERT Exemplar Solutions of Class 10 Subject Wise

Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.

Frequently Asked Questions (FAQs)

Q: According to NCERT solutions for Class 10 Maths chapter 1 exercise 1.2 , What is the Fundamental Theorem of Arithmetic ?
A:

Theorem: Every composite number may be factored as a product of primes, which means that any composite number can be expressed as a product of primes

Q: What kinds of questions do the NCERT solutions for Class 10 Maths chapter 1 exercise 1.2 cover ?
A:

The questions are based on the concept of the Fundamental Theorem of Arithmetic, in finding HCF and LCM, questions related to the relationship of HCF and LCM.

Q: Write the prime factors of 144
A:

The prime factors of 144,

144 = 2 × 2 × 2 × 2 × 3 × 3

Students can practice problems discussed in class 10 ex 1.2 to command these concepts.

Q: Is 0 a Composite Number?
A:

Zero is neither prime nor a composite number because it does not have any prime factors.

Q: What is the full form of HCF of LCM in Maths?
A:

The full form of HCF is “Highest Common Factor” and LCM is  “Least common multiple”.

Q: What are the methods used in finding LCM and HCF?
A:

The methods used in finding LCM and HCF are the prime factorization method and long division method. 

Q: Find the highest common factor of 104 and 160
A:

104 = 2 × 2 × 2 × 13

160 = 2 × 2 × 2 × 2 × 2 × 5

The common factors of 104 and 160 are 2 × 2 × 2 = 8

Therefore, HCF (104, 160) = 8

Q: Find the least common multiple of 21 and 28 .
A:

21=3×7 

28=4×7 

Now, LCM (21, 28)=7×4×3=84

Q: Write the formula which involves both HCF and LCM.
A:

The formula that involves both HCF and LCM is 

(HCF of the two numbers) x (LCM of the two numbers)= Product of Two numbers 

Q: The HCF of two numbers is 7 and their LCM is 84 . If one of the numbers is 21 then find the other number.
A:

Let the other number be a. 

We know that, 

(HCF of the two numbers) x (LCM of the two numbers)= Product of Two numbers 

Now, 7×84=21×a

 588=21a

Which on dividing both sides by 21 , 

We get, a=28 

The other number is 28 . 

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