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Real Numbers

Madhya Pradesh Board · Class 10 · Mathematics

NCERT Solutions for Real Numbers — Madhya Pradesh Board Class 10 Mathematics.

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A Venn diagram illustrating the classification of real numbers into rational and irrational numbers, with further subdivisions for rational numbers (integers, whole numbers, natural numbers).
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10 Questions Solved · 2 Sections

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EXERCISE 1.1

1Express each number as a product of its prime factors:Show solution
Prime factorisation:

- (i)  140=2×2×5×7=22×5×7(i)\;140 = 2 \times 2 \times 5 \times 7 = 2^2 \times 5 \times 7
- (ii)  156=2×2×3×13=22×3×13(ii)\;156 = 2 \times 2 \times 3 \times 13 = 2^2 \times 3 \times 13
- (iii)  3825=3×3×5×11×23=32×5×11×23(iii)\;3825 = 3 \times 3 \times 5 \times 11 \times 23 = 3^2 \times 5 \times 11 \times 23
- (iv)  5005=5×7×11×13(iv)\;5005 = 5 \times 7 \times 11 \times 13
- (v)  7429=17×19×23(v)\;7429 = 17 \times 19 \times 23

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2Find the LCM and HCF of the following pairs of integers and verify that LCM×HCF=\text{LCM} \times \text{HCF} = product of the two numbers.Show solution
Using prime factorisation:

- (i)  26=2×13(i)\;26 = 2 \times 13, 91=7×1391 = 7 \times 13
- HCF =13= 13
- LCM =2×7×13=182= 2 \times 7 \times 13 = 182
- Check: 13×182=2366=26×9113 \times 182 = 2366 = 26 \times 91

- (ii)  510=2×3×5×17(ii)\;510 = 2 \times 3 \times 5 \times 17, 92=22×2392 = 2^2 \times 23
- HCF =2= 2
- LCM =22×3×5×17×23=23460= 2^2 \times 3 \times 5 \times 17 \times 23 = 23460
- Check: 2×23460=46920=510×922 \times 23460 = 46920 = 510 \times 92

- (iii)  336=24×3×7(iii)\;336 = 2^4 \times 3 \times 7, 54=2×3354 = 2 \times 3^3
- HCF =21×31=6= 2^1 \times 3^1 = 6
- LCM =24×33×7=1512= 2^4 \times 3^3 \times 7 = 1512
- Check: 6×1512=9072=336×546 \times 1512 = 9072 = 336 \times 54

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3Find the LCM and HCF of the following integers by applying the prime factorisation method.Show solution
Using prime factorisation:

- (i)  12=22×3(i)\;12 = 2^2 \times 3, 15=3×515 = 3 \times 5, 21=3×721 = 3 \times 7
- HCF =3= 3
- LCM =22×3×5×7=420= 2^2 \times 3 \times 5 \times 7 = 420

- (ii)  17,23,29(ii)\;17, 23, 29 are all primes and have no common factor other than 1
- HCF =1= 1
- LCM =17×23×29=11339= 17 \times 23 \times 29 = 11339

- (iii)  8=23(iii)\;8 = 2^3, 9=329 = 3^2, 25=5225 = 5^2
- HCF =1= 1
- LCM =23×32×52=1800= 2^3 \times 3^2 \times 5^2 = 1800

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4Given that HCF (306,657)=9\text{HCF } (306, 657) = 9, find LCM (306,657)\text{LCM } (306, 657).Show solution
Use the relation

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

So,

LCM(306,657)=306×6579\text{LCM}(306,657)=\frac{306\times657}{9}

First calculate:

306×657=201042306\times657 = 201042

Now divide by 99:

2010429=22338\frac{201042}{9}=22338

So the correct LCM is 22338. This is the computed value; if a different value appears elsewhere, it is not correct.

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5Check whether 6n6^n can end with the digit 0 for any natural number nn.Show solution
No. If 6n6^n ended with 00, it would be divisible by 55. But

6n=(2×3)n6^n = (2\times 3)^n

so its prime factors are only 22 and 33. There is no factor 55 in its factorisation. Therefore, by the Fundamental Theorem of Arithmetic, 6n6^n cannot end with the digit 00 for any natural number nn.

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6Explain why 7×11×13+137 \times 11 \times 13 + 13 and 7×6×5×4×3×2×1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 are composite numbers.
7There 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

EXERCISE 1.2

1Prove that 5\sqrt{5} is irrational.
2Prove that 3+253 + 2\sqrt{5} is irrational.
3Prove that the following are irrationals :

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Frequently Asked Questions

What are the important topics in Real Numbers for Madhya Pradesh Board Class 10 Mathematics?
Key topics in Real Numbers include How to Correctly Identify Rational vs Irrational Numbers, Real Numbers Chapter — Complete Concept Map, Real Numbers — Complete Chapter Overview. These are the concepts Madhya Pradesh Board Class 10 examiners draw on most — study them first, then practise related questions.
How to score full marks in Real Numbers — Madhya Pradesh Board Class 10 Mathematics?
Understand the core concepts first, then work through the 116 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
Where can I get free NCERT Solutions for Real Numbers Class 10 Mathematics?
This page has free step-by-step NCERT Solutions for every exercise question in Real Numbers (Madhya Pradesh Board Class 10 Mathematics) — written the way examiners award marks: given, formula, working, answer.

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