NCERT Class 10 Maths – Chapter 1: Real Numbers
Exercise 1.1 – Complete Solutions with Explanation
Chapter: Real Numbers
Class: 10
Book: NCERT
Exercise: 1.1
Question 1: Express each number as a product of its prime factors.
(i) 140
140 = 2 × 70
= 2 × 2 × 35
= 2 × 2 × 5 × 7
Prime Factorisation: 140 = 2² × 5 × 7
(ii) 156
156 = 2 × 78
= 2 × 2 × 39
= 2 × 2 × 3 × 13
Prime Factorisation: 156 = 2² × 3 × 13
(iii) 3825
3825 = 5 × 765
= 5 × 5 × 153
= 5 × 5 × 3 × 51
= 5 × 5 × 3 × 3 × 17
Prime Factorisation: 3825 = 3² × 5² × 17
(iv) 5005
5005 = 5 × 1001
= 5 × 7 × 143
= 5 × 7 × 11 × 13
Prime Factorisation: 5005 = 5 × 7 × 11 × 13
(v) 7429
7429 = 17 × 437
= 17 × 19 × 23
Prime Factorisation: 7429 = 17 × 19 × 23
Question 2: Find the LCM and HCF and verify that LCM × HCF = Product of the two numbers.
(i) 26 and 91
26 = 2 × 13
91 = 7 × 13
HCF: 13
LCM: 2 × 7 × 13 = 182
Verification:
LCM × HCF = 182 × 13 = 2366
Product of numbers = 26 × 91 = 2366 ✔
(ii) 510 and 92
510 = 2 × 3 × 5 × 17
92 = 2² × 23
HCF: 2
LCM: 2² × 3 × 5 × 17 × 23 = 23460
Verification:
LCM × HCF = 23460 × 2 = 46920
Product = 510 × 92 = 46920 ✔
(iii) 336 and 54
336 = 2⁴ × 3 × 7
54 = 2 × 3³
HCF: 2 × 3 = 6
LCM: 2⁴ × 3³ × 7 = 3024
Verification:
LCM × HCF = 3024 × 6 = 18144
Product = 336 × 54 = 18144 ✔
Question 3: Find the LCM and HCF of the following integers by prime factorisation method.
(i) 12, 15 and 21
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7
HCF: 3
LCM: 2² × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
All are prime numbers.
HCF: 1
LCM: 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
8 = 2³
9 = 3²
25 = 5²
HCF: 1
LCM: 2³ × 3² × 5² = 1800
Question 4: Given that HCF (306, 657) = 9, find LCM (306, 657).
We use the identity:
LCM × HCF = Product of the two numbers
LCM = (306 × 657) / 9
LCM = 200, 742 / 9
LCM = 22338
Question 5: Check whether 6ⁿ can end with the digit 0 for any natural number n.
A number ends with 0 if it is divisible by 10.
10 = 2 × 5
6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ
There is no factor 5 in 6ⁿ.
Therefore, 6ⁿ is not divisible by 10.
Hence, 6ⁿ cannot end with the digit 0 for any natural number n.
Question 6: Show that the given numbers are composite.
(i) 7 × 11 × 13 + 13
= 13(7 × 11 + 1)
= 13(77 + 1)
= 13 × 78
= 1014
Since it has factors other than 1 and itself, it is composite.
(ii) 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= 7! + 5
= 5040 + 5
= 5045
= 5 × 1009
Therefore, it is composite.
Question 7: Circular Path Problem
Sonia takes 18 minutes per round.
Ravi takes 12 minutes per round.
Time when both meet again = LCM (18, 12)
18 = 2 × 3²
12 = 2² × 3
LCM = 2² × 3² = 36
They will meet again after 36 minutes.
Important Concepts Used
- Prime Factorisation Method
- HCF (Highest Common Factor)
- LCM (Least Common Multiple)
- Fundamental Theorem of Arithmetic
- Relation: LCM × HCF = Product of two numbers