In Class 10 Math Ch 1 Real Numbers Ex 1.1 begins with the fundamental concept of Real Numbers, forming the foundation for many topics ahead. Exercise 1.1 mainly deals with the Euclid’s Division Lemma, a powerful method used to find the highest common factor (HCF) of two positive integers.
Class 10 Math Ch 1 Real Numbers Ex 1.1 – Textbook
Express each number as a product of its prime factors:
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Solution:
(i) 140
140 = 2 × 70
= 2 × 2 × 35
= 2 × 2 × 5 × 7
Answer: 140 = 2² × 5 × 7
(ii) 156
156 = 2 × 78
= 2 × 2 × 39
= 2 × 2 × 3 × 13
Answer: 156 = 2² × 3 × 13
(iii) 3825
3825 = 5 × 765
= 5 × 5 × 153
= 5 × 5 × 3 × 51
= 5 × 5 × 3 × 3 × 17
Answer: 3825 = 3² × 5² × 17
(iv) 5005
5005 = 5 × 1001
= 5 × 7 × 143
= 5 × 7 × 11 × 13
Answer: 5005 = 5 × 7 × 11 × 13
(v) 7429
7429 = 17 × 437
= 17 × 19 × 23
Answer: 7429 = 17 × 19 × 23
2. 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
Solution
(i) 26 and 91
Prime factors:
26 = 2 × 13
91 = 7 × 13
HCF (common factors):
HCF = 13
LCM (all factors):
LCM = 2 × 7 × 13 = 182
Verification:
LCM × HCF = 182 × 13 = 2366
Product = 26 × 91 = 2366 ✅
(ii) 510 and 92
Prime factors:
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23 = 2² × 23
HCF (common factors):
HCF = 2
LCM (highest powers):
LCM = 2² × 3 × 5 × 17 × 23 = 46920
Verification:
LCM × HCF = 46920 × 2 = 93840
Product = 510 × 92 = 93840 ✅
(iii) 336 and 54
Prime factors:
336 = 2⁴ × 3 × 7
54 = 2 × 3³
HCF (lowest powers):
HCF = 2 × 3 = 6
LCM (highest powers):
LCM = 2⁴ × 3³ × 7 = 3024
Verification:
LCM × HCF = 3024 × 6 = 18144
Product = 336 × 54 = 18144 ✅
Ex 1.1 Class 10 Maths Question 3.
Find the LCM and HCF of the following integers by applying the prime factorisation
method.
(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
Solution:
(i) 12, 15 and 21
Prime factors:
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7
HCF (common factors in all):
HCF = 3
LCM (highest powers of all primes):
LCM = 2² × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
All are prime numbers.
HCF:
No common factor except 1 → HCF = 1
LCM:
LCM = 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
Prime factors:
8 = 2³
9 = 3²
25 = 5²
HCF:
No common factor → HCF = 1
LCM:
LCM = 2³ × 3² × 5² = 1800
Ex 1.1 Class 10 Maths Question 4.
Given that HCF (306, 657) = 9, find LCM (306, 657).
Solution:
Use the formula:
LCM × HCF = Product of the numbers
So,
LCM = (306 × 657) ÷ 9
First multiply:
306 × 657 = 201042
Now divide:
LCM = 201042 ÷ 9 = 22338
Ex 1.1 Class 10 Maths Question 5.
Check whether 6n
can end with the digit 0 for any natural number n.
Solution:
A number ends with 0 only if it is divisible by 10.
And
10 = 2 × 5
So, the number must have both 2 and 5 as factors.
Now consider :
So,
Observation:
- It contains factor 2
- It contains factor 3
- It does NOT contain factor 5
Since does not have factor 5, it can never be divisible by 10.
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Sol:
(i) 7 × 11 × 13 + 13
Take 13 common:
7 × 11 × 13 + 13
= 13(7 × 11 + 1)
= 13(77 + 1)
= 13 × 78
Now,
78 = 2 × 3 × 13
So the number = 13 × 78 = 13 × 2 × 3 × 13
👉 It has more than two factors ⇒ Composite number
(ii) 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
First write:
= 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
Take 5 common:
= 5(7 × 6 × 4 × 3 × 2 × 1 + 1)
= 5(1008 + 1)
= 5 × 1009
Now,
1009 = 7 × 144 + 1 (not divisible by 7, but still >1)
So the number = 5 × 1009
👉 It has factors 5 and 1009 ⇒ Composite number
7. 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?
Solution:
- Sonia takes = 18 minutes (one round)
- Ravi takes = 12 minutes (one round)
They will meet at the starting point when both complete whole number of rounds together, i.e., at the LCM of 18 and 12.
18 = 2 × 3²
12 = 2² × 3
LCM = 2² × 3² = 4 × 9 = 3
They will meet again at the starting point after 36 minutes.
Class 10 Math Ch 1 Real Numbers Ex 1.1
For additional reference and to access the official NCERT Class 10 Maths textbook, visit the NCERT website. This will help you understand the concepts covered in Class 10 Math Ch 1 Real Numbers Ex 1.1 more effectively.
🔗 Visit NCERT Official Website: NCERT Class 10 Maths Book
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Understanding the foundation of real numbers is essential for mastering higher-level mathematics. On this page, we have provided Class 10 Math Ch 1 Real Numbers Ex 1.1 all solution to help students grasp the concepts clearly and thoroughly. These exercises cover important topics like Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and rational and irrational numbers.
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Whether you’re revising for exams or learning these concepts for the first time, our comprehensive guide to Class 10 Math Ch 1 Real Numbers Ex 1.1 offers the support you need. With regular practice and the right guidance, success in mathematics becomes much more achievable.