NCERT Solutions Class 10 Maths Chapter 1: Real Numbers

Step-by-step solutions for Exercise 1.1 (Fundamental Theorem of Arithmetic).

Ex 1.1 Ex 1.2
Exercise 1.1 • Q1

Express each number as a product of its prime factors:
(i) 140   (ii) 156   (iii) 3825   (iv) 5005   (v) 7429

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Solution (i) 140
Divide by the smallest prime numbers sequentially:
140 = 2 × 70
= 2 × 2 × 35
= 2 × 2 × 5 × 7
Answer: 2² × 5 × 7
Solution (ii) 156
156 = 2 × 78
= 2 × 2 × 39
= 2 × 2 × 3 × 13
Answer: 2² × 3 × 13
Solution (iii) 3825
Since the number ends in 5, it is divisible by 5, but sum of digits (3+8+2+5=18) is divisible by 9, so start with 3:
3825 = 3 × 1275
= 3 × 3 × 425
= 3 × 3 × 5 × 85
= 3 × 3 × 5 × 5 × 17
Answer: 3² × 5² × 17
Solution (iv) 5005
5005 = 5 × 1001
= 5 × 7 × 143
= 5 × 7 × 11 × 13
Answer: 5 × 7 × 11 × 13
Solution (v) 7429
This requires checking larger primes:
7429 ÷ 17 = 437
437 ÷ 19 = 23
Answer: 17 × 19 × 23
Exercise 1.1 • Q2

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the two numbers.
(i) 26 and 91   (ii) 510 and 92   (iii) 336 and 54

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(i) 26 and 91
Step 1: Prime Factorization
26 = 2 × 13
91 = 7 × 13

Step 2: Find HCF and LCM
HCF = Common factor = 13
LCM = Product of greatest powers = 2 × 7 × 13 = 182

Step 3: Verification
Product of numbers = 26 × 91 = 2366
HCF × LCM = 13 × 182 = 2366
Hence Verified.
(ii) 510 and 92
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23 = 2² × 23

HCF = 2 (Lowest power of common factor)
LCM = 2² × 3 × 5 × 17 × 23 = 23460

Verification:
510 × 92 = 46920
2 × 23460 = 46920 (Verified)
(iii) 336 and 54
336 = 2⁴ × 3 × 7
54 = 2 × 3³

HCF = 2 × 3 = 6
LCM = 2⁴ × 3³ × 7 = 3024

Verification:
336 × 54 = 18144
6 × 3024 = 18144 (Verified)

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Exercise 1.1 • Q3

Find the LCM and HCF of the following integers by applying the prime factorization method.
(i) 12, 15 and 21   (ii) 17, 23 and 29   (iii) 8, 9 and 25

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(i) 12, 15 and 21
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7

HCF (Common factor with lowest power) = 3
LCM (All factors with highest power) = 2² × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
These are all prime numbers.
HCF = 1
LCM = 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
8 = 2³
9 = 3²
25 = 5²
There are no common factors other than 1.
HCF = 1
LCM = 8 × 9 × 25 = 1800
Exercise 1.1 • Q4

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

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Step 1: Use the Formula
We know that: LCM × HCF = Product of Numbers
Step 2: Substitute Values
LCM × 9 = 306 × 657
LCM = (306 × 657) / 9
Step 3: Calculate
Divide 306 by 9 = 34
LCM = 34 × 657
LCM = 22338
Exercise 1.1 • Q5

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

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Logic
For a number to end with the digit 0, it must be divisible by 10. This means its prime factorization must contain both 2 and 5.
Analysis of 6ⁿ
The prime factorization of 6 is 2 × 3.
Therefore, 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ.
Conclusion
The factorization contains 2 but does not contain 5.
By the Fundamental Theorem of Arithmetic, this factorization is unique.
Hence, 6ⁿ can never end with the digit 0 for any natural number n.

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Exercise 1.1 • Q6

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

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Definition
Composite numbers are numbers that have factors other than 1 and themselves. We need to prove these numbers have common factors.
Part 1
Expression: 7 × 11 × 13 + 13
Take 13 common:
= 13 × (7 × 11 + 1)
= 13 × (77 + 1)
= 13 × 78
Since this number has factors 13 and 78 (other than 1), it is a composite number.
Part 2
Expression: 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
Take 5 common:
= 5 × [(7 × 6 × 4 × 3 × 2 × 1) + 1]
= 5 × [1008 + 1]
= 5 × 1009
Since it has factors 5 and 1009, it is a composite number.
Exercise 1.1 • Q7

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round, while Ravi takes 12 minutes. 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?

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Step 1: Identify the Concept
They will meet again when both have completed a whole number of rounds. This time must be a common multiple of 18 and 12. We need the LCM (Lowest Common Multiple).
Step 2: Prime Factorization
18 = 2 × 3²
12 = 2² × 3
Step 3: Find LCM
LCM = Product of greatest power of each prime factor.
LCM = 2² × 3² = 4 × 9 = 36
Answer
They will meet again at the starting point after 36 minutes.