NCERT Solutions – CBSE Class 10 Mathematics

In this post, we have provided complete step-by-step solutions for Exercise 1.1 Real Numbers from the NCERT Class 10 Maths textbook. All questions are solved in an easy and exam-oriented format for better understanding.

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Quick Information

Item	Details
Board	CBSE
Class	10
Subject	Maths
Chapter	Real Numbers
Exercise	1.1

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Concepts Used

• Prime Factorisation
• HCF & LCM
• Fundamental Theorem of Arithmetic
• Divisibility Rules

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Important Formula

$\text{HCF}(a,b) \times \text{LCM}(a,b)=a\times b$HCF(a,b)×LCM(a,b)=a×b

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Question 1

Express each number as a product of its prime factors.

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(i) 140

Solution

Prime factorisation of 140:

140 = 2 × 70
70 = 2 × 35
35 = 5 × 7

Therefore,

140 = 2 × 2 × 5 × 7

Final Answer:

140 = 2² × 5 × 7

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(ii) 156

Solution

156 = 2 × 78
78 = 2 × 39
39 = 3 × 13

Therefore,

156 = 2 × 2 × 3 × 13

Final Answer:

156 = 2² × 3 × 13

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(iii) 3825

Solution

3825 = 5 × 765
765 = 5 × 153
153 = 3 × 51
51 = 3 × 17

Therefore,

3825 = 5 × 5 × 3 × 3 × 17

Final Answer:

3825 = 3² × 5² × 17

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(iv) 5005

Solution

5005 = 5 × 1001
1001 = 7 × 143
143 = 11 × 13

Therefore,

5005 = 5 × 7 × 11 × 13

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(v) 7429

Solution

7429 = 17 × 437
437 = 19 × 23

Therefore,

7429 = 17 × 19 × 23

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Question 2

Find the LCM and HCF of the following pairs of integers and verify that:

$\text{LCM} \times \text{HCF}=\text{Product of the two numbers}$LCM×HCF=Product of the two numbers

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(i) 26 and 91

Solution

Prime factorisation:

26 = 2 × 13
91 = 7 × 13

HCF:

HCF = 13

LCM:

LCM = 2 × 7 × 13 = 182

Verification:

13 × 182 = 2366
26 × 91 = 2366

Hence verified.

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(ii) 510 and 92

Solution

Prime factorisation:

510 = 2 × 3 × 5 × 17
92 = 2² × 23

HCF:

HCF = 2

LCM:

LCM = 2² × 3 × 5 × 17 × 23
= 23460

Verification:

23460 × 2 = 46920
510 × 92 = 46920

Hence verified.

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(iii) 336 and 54

Solution

Prime factorisation:

336 = 2⁴ × 3 × 7
54 = 2 × 3³

HCF:

HCF = 2 × 3 = 6

LCM:

LCM = 2⁴ × 3³ × 7
= 3024

Verification:

3024 × 6 = 18144
336 × 54 = 18144

Hence verified.

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Question 3

Find the LCM and HCF using prime factorisation method.

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(i) 12, 15 and 21

Solution

Prime factorisation:

12 = 2² × 3
15 = 3 × 5
21 = 3 × 7

HCF:

HCF = 3

LCM:

LCM = 2² × 3 × 5 × 7
= 420

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(ii) 17, 23 and 29

Solution

All numbers are prime.

HCF:

HCF = 1

LCM:

LCM = 17 × 23 × 29
= 11339

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(iii) 8, 9 and 25

Solution

Prime factorisation:

8 = 2³
9 = 3²
25 = 5²

HCF:

HCF = 1

LCM:

LCM = 2³ × 3² × 5²
= 1800

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Question 4

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

Solution

Using formula:

$\text{LCM}=\frac{a\times b}{\text{HCF}}$LCM=HCFa×b​

Substituting values:

LCM = (306 × 657) / 9

= 22338

Final Answer:

LCM = 22338

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Question 5

Check whether 6ⁿ can end with digit 0 for any natural number n.

Solution

We know:

6 = 2 × 3

Therefore:

6ⁿ = 2ⁿ × 3ⁿ

A number ending with 0 must contain factor 5.

But prime factorisation of 6ⁿ contains only 2 and 3.

Therefore, 6ⁿ can never end with digit 0.

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Question 6

Explain why the following are composite numbers.

(i) 7 × 11 × 13 + 13

Solution

Taking 13 common:

= 13(7 × 11 + 1)

= 13(77 + 1)

= 13 × 78

Since it has factors other than 1 and itself, it is composite.

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(ii) 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5

Solution

Taking 5 common:

= 5(7 × 6 × 4 × 3 × 2 × 1 + 1)

Hence it is divisible by 5.

Therefore, it is composite.

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Question 7

Sonia takes 18 minutes and Ravi takes 12 minutes to complete one round of a sports field. After how many minutes will they meet again at the starting point?

Solution

We find LCM of 18 and 12.

Prime factorisation:

18 = 2 × 3²
12 = 2² × 3

LCM:

LCM = 2² × 3²
= 36

Final Answer:

They will meet again after 36 minutes.

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Common Mistakes

• Wrong prime factorisation
• Forgetting highest/smallest powers in LCM & HCF
• Calculation mistakes in multiplication

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Exam Tips

• Learn prime tables till 50
• Practice factorisation daily
• Verify LCM × HCF relation carefully

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FAQs

Is Exercise 1.1 important for board exams?

Yes, questions based on HCF, LCM and prime factorisation are frequently asked in CBSE exams.

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Which theorem is used in this exercise?

Fundamental Theorem of Arithmetic.

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What is the best way to solve these questions?

Use step-by-step prime factorisation carefully and verify calculations.

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Related Topics

• Real Numbers Important Questions
• Real Numbers MCQs
• Real Numbers Notes
• Exercise 1.2 Solutions