Mathematics builds logical thinking and problem-solving skills, and Chapter 1: Real Numbers lays the foundation for many important concepts in Class 10 Maths. This chapter focuses on properties of real numbers, Euclid’s Division Algorithm, HCF and LCM, prime factorisation, and proofs related to irrational numbers.
These NCERT Solutions for Class 10 Maths Chapter 1 include important CBSE board questions asked between 2020 and 2025. All solutions are explained step by step in simple language to help students understand concepts clearly and score well in board examinations.
NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers
Q.
Use Euclid’s division algorithm to find the HCF of:
- 135 and 225
- 196 and 38220
- 867 and 255
Q.
In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48,80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.
[CBSE - 2024]
Q.
Show that the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.
[CBSE - 2020]
Q.
Show that the number 5 × 11 × 17 + 3 × 11 is a composite number.
[CBSE - 2024]
Q.
Prove that
5 – 23 is an irrational number. It is given that
3 is an irrational number.
[CBSE - 2024]
Q.
Prove that
51 is an irrational number.
[CBSE - 2025]
Q.
Show that every positive odd integer is of the form (4q + 1) and (4q + 3), where q is some integer.
[CBSE - 2023]
Q.
Find the HCF of 1260 and 7344 using Euclid's algorithm.
[CBSE - 2023]
Q.
Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.
[CBSE - 2023]
Q.
Assertion (A) : The perimeter of
△ABC is a rational number.
Reason (R) : The sum of the squares of two rational numbers is always rational.
[CBSE - 2023]
Q.
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Q.
If two positive integers p and q can be expressed as
p=18 a2b4 and
q=20 a3b2, where a and b are prime numbers, then LCM(p, q) is:
[CBSE - 2024]
Q.
Which of the following is a rational number between
3 and 5?
[CBSE - 2025]
Q.
If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n – 7m is:
[CBSE - 2025]
Q.
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
Q.
Express each number as a product of its prime factors:
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Q.
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+ 8.
Q.
Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q+ 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]
Q.
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Q.
Prove that one of the every three consecutive positive integer is divisible by 3.
[CBSE - 2020]
Class 10 Maths Chapter 1 Questions & Answers – Real Numbers
Important Board Questions (CBSE 2020–2025)
Q1. (Medium) [CBSE – 2023]
Assertion (A): The perimeter of triangle ABC is a rational number.
Reason (R): The sum of the squares of two rational numbers is always rational.
Solution:
The assertion is not always true because the side lengths of a triangle can be irrational, making the perimeter irrational.
The reason is true because squares of rational numbers are rational and their sum is also rational.
Therefore, Assertion is false and Reason is true.
Q2. (Medium) [CBSE – 2025]
Prove that 1/√5 is an irrational number.
Solution:
Assume that 1/√5 is rational. Then its reciprocal √5 would also be rational.
But √5 is irrational, which leads to a contradiction.
Hence, 1/√5 is irrational.
Q3. (Medium) [CBSE – 2024]
Show that the number 5 × 11 × 17 + 3 × 11 is a composite number.
Solution:
5 × 11 × 17 + 3 × 11 = 11(5 × 17 + 3)
= 11(88) = 11 × 88
Since it is a product of two integers greater than 1, the number is composite.
Q4. (Medium) [CBSE – 2024]
Prove that 5 − 2√3 is an irrational number.
Solution:
Assume 5 − 2√3 is rational.
Then 2√3 becomes rational, implying √3 is rational.
This contradicts the given fact.
Therefore, 5 − 2√3 is irrational.
Q5. (Medium) [CBSE – 2023]
Show that every positive odd integer is of the form (4q + 1) or (4q + 3).
Solution:
Let n be any positive odd integer.
On dividing n by 4, the remainder can be 1 or 3.
Hence, n = 4q + 1 or 4q + 3.
Euclid’s Division Algorithm Based Questions
Q6. (Easy) [CBSE – 2023]
Find the HCF of 1260 and 7344 using Euclid’s algorithm.
Solution:
7344 = 1260 × 5 + 1044
1260 = 1044 × 1 + 216
1044 = 216 × 4 + 180
216 = 180 × 1 + 36
180 = 36 × 5 + 0
HCF = 36
Q7. (Easy) [CBSE – 2025]
Find a rational number between √3 and √5.
Solution:
√3 ≈ 1.732 and √5 ≈ 2.236.
The rational number 2 lies between them.
NCERT Solutions for Class 10 Maths Chapter 1 – FAQs
Q1. Why is Chapter 1 Real Numbers important for board exams?
This chapter is frequently asked in CBSE exams and forms the base for higher mathematics. Topics like Euclid’s Division Algorithm, HCF–LCM, and proofs of irrational numbers are scoring and concept-based.
Q2. Which topics from Real Numbers are most important?
- Euclid’s Division Algorithm
- HCF and LCM problems
- Proofs of irrational numbers
- Prime factorisation
Q3. How should students prepare Chapter 1 for exams?
Students should practice proof-based questions regularly, understand the steps of Euclid’s algorithm clearly, and revise solved examples to gain confidence and accuracy.