Mathematics · Textbook solutions

Real Numbers

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 27 questions

1.2 The Fundamental Theorem of Arithmetic

19 q

Solved Examples

Worked · 4

1.1 Eg.1 $4^n$
1.1 Eg.2 $Find the LCM and HCF of 6 and 20 by the prime factorisation method.$
1.1 Eg.3 $Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.$
1.1 Eg.4 $Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.$

Exercise 1.1

Practice · 15

Express each number as a product of its prime factors:
Ex 1.1 Q1 (i) $140$
Ex 1.1 Q1 (ii) $156$
Ex 1.1 Q1 (iii) $3825$
Ex 1.1 Q1 (iv) $5005$
Ex 1.1 Q1 (v) $7429$
Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers.
Ex 1.1 Q2 (i) $26 and 91$
Ex 1.1 Q2 (ii) $510 and 92$
Ex 1.1 Q2 (iii) $336 and 54$
Find the LCM and HCF of the following integers by applying the prime factorisation method.
Ex 1.1 Q3 (i) $12, 15 and 21$
Ex 1.1 Q3 (ii) $17, 23 and 29$
Ex 1.1 Q3 (iii) $8, 9 and 25$
Ex 1.1 Q4 $\text{HCF}(306, 657) = 9$
Ex 1.1 Q5 $6^n$
Ex 1.1 Q6 $7 \times 11 \times 13 + 13$
Ex 1.1 Q7 $There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?$

1.3 Revisiting Irrational Numbers

8 q

Solved Examples

Worked · 3

1.2 Eg.5 $\sqrt{3}$
1.2 Eg.6 $5 - \sqrt{3}$
1.2 Eg.7 $3\sqrt{2}$

Exercise 1.2

Practice · 5

Ex 1.2 Q1 $\sqrt{5}$
Ex 1.2 Q2 $3 + 2\sqrt{5}$
Prove that the following are irrationals :
Ex 1.2 Q3 (i) $\frac{1}{\sqrt{2}}$
Ex 1.2 Q3 (ii) $7\sqrt{5}$
Ex 1.2 Q3 (iii) $6 + \sqrt{2}$

What the board has asked from this chapter

Questions per paper · 63 papers · 2023 to 2026

2026

20 q

1 $The HCF of 960 and 432 is :$
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HCF and LCM by Prime Factorisation
1 $The LCM of 960 and 240 is :$
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HCF and LCM by Prime Factorisation
2 $There are two sections A and B of Grade X. There are 28 students in Section A and 30 students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B ?$
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HCF and LCM by Prime Factorisation
2 $The natural number 2 is :$
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Fundamental Theorem of Arithmetic
2 $The natural number 1 is :$
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Fundamental Theorem of Arithmetic
3 $n$
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Fundamental Theorem of Arithmetic
3 $5^n$
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Fundamental Theorem of Arithmetic
4 $(3 \times 11 \times 13 + 3)$
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Fundamental Theorem of Arithmetic
Question Numbers 19 and 20 are Assertion (A) and Reason (R) based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below :
20 $(\sqrt{3} + \sqrt{5})$
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Irrational Numbers and Proof by Contradiction
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below :
20 $(36m^2, 18m) = 18m$
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HCF and LCM by Prime Factorisation
21 $Find the length of the plank that can be used to measure the lengths 4 m 20 cm and 5 m 4 cm exactly, in the least time.$
HCF and LCM by Prime Factorisation
21 $2 - 5\sqrt{3}$
Irrational Numbers and Proof by Contradiction
23 $4 - 2\sqrt{5}$
Irrational Numbers and Proof by Contradiction
24 (a) $2 + 3\sqrt{5}$
Irrational Numbers and Proof by Contradiction
24 (b) $210 \times 5 + 55m$
HCF and LCM by Prime Factorisation
25 $14 - 2\sqrt{3}$
Irrational Numbers and Proof by Contradiction
26 $l$
HCF and LCM by Prime Factorisation
28 $Find the greatest number which divides 764 and 1198, leaving remainders 8 and 10 respectively.$
HCF and LCM by Prime Factorisation
31 $Find the greatest number less than 10,000 which is exactly divisible by 48, 60 and 65.$
HCF and LCM by Prime Factorisation
31 $\times$
HCF and LCM by Prime Factorisation

2025

36 q

1 $x = ab^3$
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HCF and LCM by Prime Factorisation
1 $\left(\sqrt{3} + 2\right)^2 + \left(\sqrt{3} - 2\right)^2$
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Irrational Numbers and Proof by Contradiction
1 $\sqrt{0.4}$
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Irrational Numbers and Proof by Contradiction
2 $x = a^2 b^3 c^n$
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HCF and LCM by Prime Factorisation
2 $If x is the LCM of 4, 6, 8 and y is the LCM of 3, 5, 7 and p is the LCM of x and y, then which of the following is true ?$
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HCF and LCM by Prime Factorisation
2 $8^n$
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Fundamental Theorem of Arithmetic
2 $\left(1 + \sqrt{3}\right)^2 - \left(1 - \sqrt{3}\right)^2$
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Irrational Numbers and Proof by Contradiction
3 $a^2$
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Irrational Numbers and Proof by Contradiction
4 $a^b = 32$
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Fundamental Theorem of Arithmetic
4 $The least number which is a perfect square and is divisible by each of 16, 20 and 50, is :$
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HCF and LCM by Prime Factorisation
4 $a = p^2 q^3 r^n$
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HCF and LCM by Prime Factorisation
6 $n - 7m$
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HCF and LCM by Prime Factorisation
9 $(-1)^n + (-1)^8 = 0$
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Fundamental Theorem of Arithmetic
10 $The sum of the exponents of prime factors in the prime factorisation of 4004 is :$
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Fundamental Theorem of Arithmetic
11 $p = x^2 y^3 z^n$
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HCF and LCM by Prime Factorisation
12 $1080 = 2^p \times 3^q \times 5$
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Fundamental Theorem of Arithmetic
14 $\sqrt{3}$
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Irrational Numbers and Proof by Contradiction
14 $The greatest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is :$
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HCF and LCM by Prime Factorisation
18 $The HCF of 40, 110 and 360 is :$
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HCF and LCM by Prime Factorisation
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from the following :
19 $p + q$
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HCF and LCM by Prime Factorisation
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from the following :
19 $x$
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HCF and LCM by Prime Factorisation
In question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option :
19 $4^n$
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Fundamental Theorem of Arithmetic
In question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option :
20 $3^n$
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Fundamental Theorem of Arithmetic
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from the following :
20 $x$
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HCF and LCM by Prime Factorisation
23 (a) $Find the smallest number which is divisible by both 644 and 462.$
HCF and LCM by Prime Factorisation
23 (b) $Two numbers are in the ratio 4 : 5 and their HCF is 11. Find the LCM of these numbers.$
HCF and LCM by Prime Factorisation
26 (a) $\sqrt{5}$
Irrational Numbers and Proof by Contradiction
26 (a) $\sqrt{2}$
Irrational Numbers and Proof by Contradiction
26 $\left(4\sqrt{2} + \dfrac{5}{3}\right)$
Irrational Numbers and Proof by Contradiction
26 (b) $p = x^2y^3$
HCF and LCM by Prime Factorisation
27 $Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of Physics books is 144, the number of Chemistry books is 180 and the number of Mathematics books is 192. Assuming that the books are of same thickness, determine the number of stacks of Physics, Chemistry and Mathematics books.$
HCF and LCM by Prime Factorisation
26 (b) $2 \times 3 \times 5 \times 7 + 7$
Fundamental Theorem of Arithmetic
26 (b) $p \cdot q \cdot r + q$
Fundamental Theorem of Arithmetic
27 $\left(5\sqrt{3} + \dfrac{2}{3}\right)$
Irrational Numbers and Proof by Contradiction
29 $\dfrac{1}{\sqrt{5}}$
Irrational Numbers and Proof by Contradiction
30 $\sqrt{3}$
Irrational Numbers and Proof by Contradiction

2024

26 q

1 $The LCM of the smallest prime number and the smallest odd composite number is :$
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HCF and LCM by Prime Factorisation
2 $The HCF of two numbers 65 and 104 is 13. If LCM of 65 and 104 is 40x, then the value of x is :$
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HCF and LCM by Prime Factorisation
5 $p = 18\,a^2b^4$
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HCF and LCM by Prime Factorisation
5 $(2520, 6600) = 40$
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HCF and LCM by Prime Factorisation
5 $a = 2^2 \times 3^x$
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HCF and LCM by Prime Factorisation
10 $The greatest number which divides 281 and 1249, leaving remainder 5 and 7 respectively, is :$
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HCF and LCM by Prime Factorisation
12 $2^3 \times 3^a \times b \times 7$
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Fundamental Theorem of Arithmetic
13 $(2520, 6600) = 40$
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HCF and LCM by Prime Factorisation
13 $The LCM of three numbers 28, 44, 132 is :$
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HCF and LCM by Prime Factorisation
14 $A pair of irrational numbers whose product is a rational number is :$
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Irrational Numbers and Proof by Contradiction
14 $If the product of two co-prime numbers is 553, then their HCF is :$
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HCF and LCM by Prime Factorisation
15 $\sqrt{20}$
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Irrational Numbers and Proof by Contradiction
16 $3825 = 3^x \times 5^y \times 17^z$
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Fundamental Theorem of Arithmetic
21 $(15)^n$
Fundamental Theorem of Arithmetic
21 $Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after what time will they next toll together ?$
HCF and LCM by Prime Factorisation
23 $In a school, there are two sections of class X. There are 40 students in the first section and 48 students in the second section. Determine the minimum number of books required for their class library so that they can be distributed equally among students of both sections.$
HCF and LCM by Prime Factorisation
25 (A) $5 - 2\sqrt{3}$
Irrational Numbers and Proof by Contradiction
25 (B) $5 \times 11 \times 17 + 3 \times 11$
Fundamental Theorem of Arithmetic
25 $8^n$
Fundamental Theorem of Arithmetic
26 (b) $\left(\sqrt{2} + \sqrt{3}\right)^2$
Irrational Numbers and Proof by Contradiction
27 $In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.$
HCF and LCM by Prime Factorisation
30 $\dfrac{2 - \sqrt{3}}{5}$
Irrational Numbers and Proof by Contradiction
Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250. Now, Mukta asked some questions as given below to the students :
38 (i) $What is the least prime number used by students ?$
Fundamental Theorem of Arithmetic
Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250. Now, Mukta asked some questions as given below to the students :
38 (ii) (a) $How many students are in the class ?$
Fundamental Theorem of Arithmetic
Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250. Now, Mukta asked some questions as given below to the students :
38 (ii) (b) $What is the highest prime number used by students ?$
Fundamental Theorem of Arithmetic
Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250. Now, Mukta asked some questions as given below to the students :
38 (iii) $Which prime number has been used maximum times ?$
Fundamental Theorem of Arithmetic

2023

23 q

1 $p^2 = \dfrac{32}{50}$
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Irrational Numbers and Proof by Contradiction
1 $If ‘n’ is a natural number, then which of the following numbers end with zero ?$
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Fundamental Theorem of Arithmetic
1 $The ratio of HCF to LCM of the least composite number and the least prime number is :$
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HCF and LCM by Prime Factorisation
4 $If ‘p’ and ‘q’ are natural numbers and ‘p’ is the multiple of ‘q’, then what is the HCF of ‘p’ and ‘q’ ?$
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HCF and LCM by Prime Factorisation
6 $The LCM of smallest 2-digit number and smallest composite number is$
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HCF and LCM by Prime Factorisation
In Question 19 and 20, an Assertion (A) statement is followed by a statement of Reason (R). Select the correct option out of the following :
20 $\Delta ABC$
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Irrational Numbers and Proof by Contradiction
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below.
20 $5^n$
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Fundamental Theorem of Arithmetic
21 $Find the greatest 3-digit number which is divisible by 18, 24 and 36.$
HCF and LCM by Prime Factorisation
21 $Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers ?$
HCF and LCM by Prime Factorisation
21 $Find the least number which when divided by 12, 16 and 24 leaves remainder 7 in each case.$
HCF and LCM by Prime Factorisation
21 $2 + \sqrt{3}$
Irrational Numbers and Proof by Contradiction
21 (A) $6^n$
Fundamental Theorem of Arithmetic
21 (B) $Find the HCF and LCM of 72 and 120.$
HCF and LCM by Prime Factorisation
23 $Using prime factorisation, find HCF and LCM of 96 and 120.$
HCF and LCM by Prime Factorisation
23 (B) $Find the LCM and HCF of 72 and 120.$
HCF and LCM by Prime Factorisation
23 (B) $Find the LCM and HCF of 72 and 120$
HCF and LCM by Prime Factorisation
24 $Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.$
HCF and LCM by Prime Factorisation
25 $4^n$
Fundamental Theorem of Arithmetic
25 $6 - \sqrt{7}$
Irrational Numbers and Proof by Contradiction
26 (a) $Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.$
HCF and LCM by Prime Factorisation
26 (b) $The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change together next ?$
HCF and LCM by Prime Factorisation
27 (A) $Find the HCF and LCM of 26, 65 and 117, using prime factorisation.$
HCF and LCM by Prime Factorisation
26 (b) $Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a.m., when will they ring together again ?$
HCF and LCM by Prime Factorisation

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 4 times: 2023*, 2024*, 2025*, 2026 (* = numbers or wording changed that year)
24 (a) $2 + 3\sqrt{5}$
Question 2
Asked 4 times: 2023*, 2024*, 2025*, 2026 (* = numbers or wording changed that year)
28 $Find the greatest number which divides 764 and 1198, leaving remainders 8 and 10 respectively.$
Question 3
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
2 $There are two sections A and B of Grade X. There are 28 students in Section A and 30 students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B ?$
Tap an option to check your answer.
Question 4
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
31 $Find the greatest number less than 10,000 which is exactly divisible by 48, 60 and 65.$
Question 5
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
21 $(15)^n$
Question 6
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
21 $Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after what time will they next toll together ?$
Question 7
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
1 $The LCM of the smallest prime number and the smallest odd composite number is :$
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Question 8
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
1 $The HCF of 960 and 432 is :$
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Question 9
This type asked 3 times: 2023, 2025, 2026. This one: 2026
3 $n$
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Question 10
This type asked 3 times: 2024, 2025, 2026. This one: 2026
4 $(3 \times 11 \times 13 + 3)$
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1.2 The Fundamental Theorem of Arithmetic

Solved Examples

Exercise 1.1

1.3 Revisiting Irrational Numbers

Solved Examples

Exercise 1.2

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10