NCERT Class X Mathematics Chapter 1: Real Numbers

Chapter 1: Real Numbers

Comprehensive Study Notes, Day-by-Day Explanations, and Question Bank

Part 1: Day-to-Day Study Notes & Step-by-Step Explanations

Day 1: The Fundamental Theorem of Arithmetic

Objective: Understand prime factorization of composite numbers and its uniqueness.

Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Step-by-Step Explanation:

• Any natural number can be written as a product of its prime factors.
• We typically arrange prime factors in ascending order (e.g., $p_1 \le p_2 \le \dots \le p_n$).
• Example: Let's factorise 32760 using a factor tree. We get $32760 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 13 = 2^3 \times 3^2 \times 5 \times 7 \times 13$.

Day 2: Finding HCF and LCM Using Prime Factorisation

Objective: Apply prime factorization to find Highest Common Factor (HCF) and Least Common Multiple (LCM), and understand their relationship.

Step-by-Step Explanation:

• HCF: Product of the smallest power of each common prime factor in the numbers.
• LCM: Product of the greatest power of each prime factor, involved in the numbers.
• Crucial Formula: For any two positive integers $a$ and $b$, $HCF(a, b) \times LCM(a, b) = a \times b$.
• Note for 3 numbers: $HCF(p, q, r) \times LCM(p, q, r) \ne p \times q \times r$.

Application Example: Check whether $4^n$ can end with the digit zero. For a number to end with zero, its prime factorisation must contain the prime 5. Since $4^n = (2)^{2n}$, the only prime factor is 2[cite: 102, 103]. Thus, it can never end with zero.

Day 3: Revisiting Irrational Numbers & Theorem 1.2

Objective: Prove the irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$ using proof by contradiction.

Theorem 1.2: Let $p$ be a prime number. If $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.

Step-by-Step Proof that $\sqrt{2}$ is irrational:

1. Assume the contrary: Let $\sqrt{2}$ be a rational number.
2. Write it in simplest form: $\sqrt{2} = \frac{a}{b}$ where $a$ and $b$ are coprime integers ($b \ne 0$)[cite: 191].
3. Rearrange and square: $b\sqrt{2} = a$ becomes $2b^2 = a^2$[cite: 192, 193].
4. This means 2 divides $a^2$, so by Theorem 1.2, 2 divides $a$. Let $a = 2c$.
5. Substitute back: $2b^2 = (2c)^2 = 4c^2 \implies b^2 = 2c^2$.
6. This means 2 divides $b^2$, so 2 divides $b$.
7. Contradiction: Both $a$ and $b$ are divisible by 2, contradicting that they are coprime[cite: 197, 199]. Thus, $\sqrt{2}$ is irrational.

Day 4: Combinations of Rational and Irrational Numbers

Objective: Prove expressions like $5 - \sqrt{3}$ are irrational.

Key Facts:

• The sum or difference of a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational and irrational number is irrational.

Step-by-Step Proof for $5 - \sqrt{3}$:

1. Assume $5 - \sqrt{3}$ is rational, meaning $5 - \sqrt{3} = \frac{a}{b}$ (coprime integers, $b \ne 0$)[cite: 231, 232, 233].
2. Rearrange to isolate the root: $\sqrt{3} = 5 - \frac{a}{b} = \frac{5b - a}{b}$.
3. Since $a$ and $b$ are integers, $\frac{5b - a}{b}$ is rational, implying $\sqrt{3}$ is rational[cite: 237, 238].
4. This contradicts the known fact that $\sqrt{3}$ is irrational. Hence, $5 - \sqrt{3}$ is irrational[cite: 239, 240].

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Part 2: Multiple Choice Questions (MCQs)

Ideal for quick conceptual testing.

1. If $HCF(306, 657) = 9$, what is the $LCM(306, 657)$?

   • 22338
   • 23328
   • 28332
   • 32238

   Answer: (a) Use formula $LCM = \frac{a \times b}{HCF}$.

2. The prime factorisation of 140 is:

   • $2 \times 7 \times 10$
   • $2^2 \times 5 \times 7$
   • $2 \times 5^2 \times 7$
   • $2^2 \times 3 \times 7$

   Answer: (b)

3. According to Theorem 1.2, if a prime $p$ divides $a^2$, then:

   • $p$ divides $a$
   • $a$ divides $p$
   • $p^2$ divides $a$
   • None of the above

   Answer: (a)

4. The HCF of 96 and 404 is:

   • 2
   • 4
   • 8
   • 12

   Answer: (b)

5. The LCM of 24 and 36 is:

   • 48
   • 60
   • 72
   • 96

   Answer: (c)

6. The prime factorisation of 180 is:

   • $2^2 \times 3^2 \times 5$
   • $2 \times 3^2 \times 5$
   • $2^2 \times 3 \times 5$
   • $2^3 \times 3^2 \times 5$

   Answer: (a)

7. If $HCF(72, 120) = 24$, then $LCM(72,120)$ is:

   • 240
   • 360
   • 480
   • 600

   Answer: (b) Use $LCM = \frac{a \times b}{HCF}$.

8. Which of the following is irrational?

   • $\sqrt{16}$
   • $\sqrt{25}$
   • $\sqrt{7}$
   • $\frac{3}{4}$

   Answer: (c)

9. The decimal expansion of $\frac{7}{8}$ is:

   • Terminating
   • Non-terminating repeating
   • Non-terminating non-repeating
   • None of these

   Answer: (a)

10. The HCF of two consecutive integers is:

    • 0
    • 1
    • 2
    • The smaller number

    Answer: (b)

11. If $p$ is a prime number and $p$ divides the product $ab$, then:

    • $p$ divides $a$ or $p$ divides $b$
    • $p$ divides $a$ and $p$ divides $b$
    • $a$ divides $p$
    • $b$ divides $p$

    Answer: (a)

12. The LCM of two coprime numbers is equal to:

    • 1
    • Their difference
    • Their product
    • Their sum

    Answer: (c)

13. The HCF of 45 and 75 is:

    • 5
    • 10
    • 15
    • 25

    Answer: (c)

14. Which of the following has a terminating decimal expansion?

    • $\frac{13}{30}$
    • $\frac{7}{64}$
    • $\frac{11}{45}$
    • $\frac{17}{27}$

    Answer: (b)

15. The prime factorisation of 225 is:

    • $3^2 \times 5^2$
    • $3^3 \times 5$
    • $9 \times 25$
    • $15^2$

    Answer: (a)

16. Euclid’s Division Lemma states that for given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that:

    • $a = bq + r$, $0 \le r < b$
    • $a = br + q$
    • $b = aq + r$
    • $a = b + q + r$

    Answer: (a)

17. The HCF of 17 and 23 is:

    • 1
    • 17
    • 23
    • 391

    Answer: (a)

18. If $LCM(a,b)=180$ and $HCF(a,b)=6$, and $a=30$, then $b$ is:

    • 18
    • 24
    • 36
    • 54

    Answer: (c)

19. The decimal expansion of $\frac{2}{11}$ is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Whole number

    Answer: (b)

20. Which of the following is a rational number?

    • $\sqrt{3}$
    • $\sqrt{2}$
    • $\frac{22}{7}$
    • $\pi$

    Answer: (c)

21. The HCF of 8, 12 and 16 is:

    • 2
    • 4
    • 8
    • 16

    Answer: (b)

22. The LCM of 8, 12 and 16 is:

    • 24
    • 32
    • 48
    • 96

    Answer: (c)

23. Which of the following fractions has a non-terminating repeating decimal?

    • $\frac{3}{25}$
    • $\frac{5}{16}$
    • $\frac{7}{15}$
    • $\frac{9}{40}$

    Answer: (c)

24. The prime factorisation of 84 is:

    • $2^2 \times 3 \times 7$
    • $2 \times 3 \times 14$
    • $4 \times 21$
    • $2^3 \times 3 \times 7$

    Answer: (a)

25. If $a$ and $b$ are coprime numbers, then $HCF(a,b)$ is:

    • 0
    • 1
    • $a$
    • $b$

    Answer: (b)

26. The LCM of 15 and 20 is:

    • 45
    • 60
    • 75
    • 90

    Answer: (b)

27. If $\sqrt{2}$ is irrational, then $5\sqrt{2}$ is:

    • Rational
    • Irrational
    • Integer
    • Whole number

    Answer: (b)

28. The HCF of 144 and 180 is:

    • 12
    • 18
    • 24
    • 36

    Answer: (d)

29. If a rational number has denominator of the form $2^m5^n$ (where $m,n$ are non-negative integers), then its decimal expansion is:

    • Terminating
    • Non-terminating repeating
    • Non-terminating non-repeating
    • Irrational

    Answer: (a)

30. The HCF of 27 and 36 using prime factorisation is:

    • 3
    • 6
    • 9
    • 12

    Answer: (c)

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Part 3: Graded Sample Question Bank

Very Short Answer Type (1 Mark)

1. Express 156 as a product of its prime factors. [1 Mark]
2. Express 84 as a product of its prime factors. [1 Mark]
3. Express 210 as a product of its prime factors. [1 Mark]
4. Express 360 as a product of its prime factors. [1 Mark]
5. Express 128 as a product of its prime factors. [1 Mark]
6. Express 945 as a product of its prime factors. [1 Mark]
7. Express 231 as a product of its prime factors. [1 Mark]
8. Express 144 as a product of its prime factors. [1 Mark]
9. Express 225 as a product of its prime factors. [1 Mark]
10. Express 396 as a product of its prime factors. [1 Mark]
11. Express 504 as a product of its prime factors. [1 Mark]
12. If $a = 2^2 \times 3^1$ and $b = 2 \times 2 \times 5$, find the $HCF(a, b)$. [1 Mark]
13. If $a = 2^3 \times 5$ and $b = 2^2 \times 3$, find the $HCF(a, b)$. [1 Mark]
14. If $a = 3^2 \times 7$ and $b = 3 \times 5^2$, find the $HCF(a, b)$. [1 Mark]
15. If $a = 2^4 \times 3$ and $b = 2^2 \times 3^2$, find the $HCF(a, b)$. [1 Mark]
16. If $a = 5^2 \times 11$ and $b = 5 \times 7$, find the $HCF(a, b)$. [1 Mark]
17. If $a = 2^2 \times 7^2$ and $b = 2 \times 7 \times 3$, find the $HCF(a, b)$. [1 Mark]
18. If $a = 3^3 \times 5$ and $b = 3^2 \times 2$, find the $HCF(a, b)$. [1 Mark]
19. If $a = 2^3 \times 11$ and $b = 2 \times 11^2$, find the $HCF(a, b)$. [1 Mark]
20. If $a = 7^2 \times 13$ and $b = 7 \times 13^2$, find the $HCF(a, b)$. [1 Mark]
21. If $a = 2^5 \times 3^2$ and $b = 2^3 \times 3$, find the $HCF(a, b)$. [1 Mark]
22. If $a = 3^2 \times 5^2$ and $b = 3 \times 5^3$, find the $HCF(a, b)$. [1 Mark]

Short Answer Type I (2 Marks)

23. Check whether $6^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
24. Check whether $4^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
25. Check whether $5^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
26. Check whether $2^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
27. Check whether $3^n$ can end with the digit 1 for any natural number $n$. [2 Marks]
28. Check whether $7^n$ can end with the digit 3 for any natural number $n$. [2 Marks]
29. Check whether $8^n$ can end with the digit 2 for any natural number $n$. [2 Marks]
30. Check whether $9^n$ can end with the digit 7 for any natural number $n$. [2 Marks]
31. Check whether $10^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
32. Check whether $12^n$ can end with the digit 5 for any natural number $n$. [2 Marks]
33. Check whether $15^n$ can end with the digit 0 for any natural number $n$. [2 Marks]
34. Prove that $3\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
35. Prove that $2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
36. Prove that $5\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
37. Prove that $4\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
38. Prove that $7\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
39. Prove that $6\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
40. Prove that $9\sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
41. Prove that $8\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
42. Prove that $10\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
43. Prove that $11\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
44. Prove that $12\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
45. Prove that $3 + 2\sqrt{5}$ is irrational. Provided that $\sqrt{5}$ is irrational.[2 Marks]
46. Prove that $4 + 3\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
47. Prove that $5 - 2\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
48. Prove that $7 + \sqrt{11}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
49. Prove that $6 - 4\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
50. Prove that $8 + 5\sqrt{13}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
51. Prove that $9 - 3\sqrt{5}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
52. Prove that $2 + 7\sqrt{3}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
53. Prove that $10 - 6\sqrt{7}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
54. Prove that $1 + 4\sqrt{6}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
55. Prove that $12 - 5\sqrt{2}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
56. Prove that $\frac{5 + 3\sqrt{5}}{4}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
57. Prove that $\frac{7 + 2\sqrt{3}}{5}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
58. Prove that $\frac{9 - 4\sqrt{2}}{3}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
59. Prove that $\frac{6 + 5\sqrt{7}}{8}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
60. Prove that $\frac{11 - 3\sqrt{5}}{6}$ is irrational, provided that $\sqrt{5}$ is irrational. [2 Marks]
61. Prove that $\frac{4 + 7\sqrt{11}}{9}$ is irrational, provided that $\sqrt{11}$ is irrational. [2 Marks]
62. Prove that $\frac{8 - 2\sqrt{13}}{7}$ is irrational, provided that $\sqrt{13}$ is irrational. [2 Marks]
63. Prove that $\frac{3 + 4\sqrt{6}}{5}$ is irrational, provided that $\sqrt{6}$ is irrational. [2 Marks]
64. Prove that $\frac{10 - 5\sqrt{3}}{12}$ is irrational, provided that $\sqrt{3}$ is irrational. [2 Marks]
65. Prove that $\frac{2 + 9\sqrt{2}}{11}$ is irrational, provided that $\sqrt{2}$ is irrational. [2 Marks]
66. Prove that $\frac{13 - 6\sqrt{7}}{10}$ is irrational, provided that $\sqrt{7}$ is irrational. [2 Marks]
67. Explain why $7 \times 11 \times 13 + 13$ is a composite number. [2 Marks]
68. Explain why $5 \times 9 \times 11 + 11$ is a composite number. [2 Marks]
69. Explain why $4 \times 7 \times 15 + 15$ is a composite number. [2 Marks]
70. Explain why $6 \times 13 \times 17 + 17$ is a composite number. [2 Marks]
71. Explain why $8 \times 9 \times 19 + 19$ is a composite number. [2 Marks]
72. Explain why $3 \times 10 \times 21 + 21$ is a composite number. [2 Marks]
73. Explain why $2 \times 5 \times 23 + 23$ is a composite number. [2 Marks]
74. Explain why $7 \times 8 \times 25 + 25$ is a composite number. [2 Marks]
75. Explain why $9 \times 14 \times 29 + 29$ is a composite number. [2 Marks]
76. Explain why $11 \times 12 \times 31 + 31$ is a composite number. [2 Marks]
77. Explain why $4 \times 6 \times 27 + 27$ is a composite number. [2 Marks]

Short Answer Type II (3 Marks)

1. Prove that $\sqrt{5}$ is an irrational number. [3 Marks]
2. Prove that $\sqrt{7}$ is an irrational number. [3 Marks]
3. Prove that $\sqrt{11}$ is an irrational number. [3 Marks]
4. Prove that $\sqrt{13}$ is an irrational number. [3 Marks]
5. Prove that $\sqrt{17}$ is an irrational number. [3 Marks]
6. Prove that $\sqrt{19}$ is an irrational number. [3 Marks]
7. Prove that $\sqrt{23}$ is an irrational number. [3 Marks]
8. Prove that $\sqrt{29}$ is an irrational number. [3 Marks]
9. Prove that $\sqrt{31}$ is an irrational number. [3 Marks]
10. Prove that $\sqrt{37}$ is an irrational number. [3 Marks]
11. Prove that $\sqrt{41}$ is an irrational number. [3 Marks]
12. Find the LCM and HCF of 240 and 96 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
13. Find the LCM and HCF of 180 and 144 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
14. Find the LCM and HCF of 210 and 84 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
15. Find the LCM and HCF of 324 and 90 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
16. Find the LCM and HCF of 150 and 225 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
17. Find the LCM and HCF of 288 and 108 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
18. Find the LCM and HCF of 420 and 168 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
19. Find the LCM and HCF of 360 and 72 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
20. Find the LCM and HCF of 252 and 198 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
21. Find the LCM and HCF of 270 and 192 by the prime factorisation method and verify that $LCM \times HCF = \text{product of the two numbers}$. [3 Marks]
22. Find the LCM and HCF of 336 and 54 by the prime factorisation method and verify that $LCM \times HCF =\text{product of the two numbers}$. [3 Marks]

Long Answer / Application Based (4 Marks)

1. 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? Explain your mathematical reasoning. [4 Marks]
2. Two runners are running around a circular track. Aman completes one round in 15 minutes, while Rohan completes one round in 20 minutes. If they start together from the same point and run in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
3. Neha and Priya walk around a circular park. Neha takes 24 minutes to complete one round, and Priya takes 16 minutes. If they start at the same time from the same point in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
4. Two cyclists take 14 minutes and 21 minutes respectively to complete one round of a circular track. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Justify your answer. [4 Marks]
5. Rahul takes 28 minutes to complete one round of a circular ground, while Mohan takes 35 minutes. If both start at the same time from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain clearly. [4 Marks]
6. Two athletes complete one round of a stadium in 18 minutes and 27 minutes respectively. If they begin together from the same point in the same direction, after how many minutes will they again reach the starting point together? Give mathematical reasoning. [4 Marks]
7. Anita and Kavya jog around a circular path. Anita takes 32 minutes for one round and Kavya takes 48 minutes. If they start simultaneously from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain your reasoning. [4 Marks]
8. Two friends complete one round of a circular park in 9 minutes and 15 minutes respectively. If they start together and move in the same direction, after how many minutes will they be together again at the starting point? Explain your method. [4 Marks]
9. Arjun and Vivek take 20 minutes and 30 minutes respectively to complete one round of a circular field. If they start at the same time from the same point and go in the same direction, after how many minutes will they meet again at the starting point? Justify your answer. [4 Marks]
10. Two walkers take 25 minutes and 40 minutes respectively to go once around a circular garden. If they begin together and walk in the same direction, after how many minutes will they reach the starting point together again? Explain clearly. [4 Marks]
11. Seema and Ritu take 36 minutes and 45 minutes respectively to complete one round of a circular path. If they start together from the same point and move in the same direction, after how many minutes will they meet again at the starting point? Explain the mathematical reasoning. [4 Marks]
12. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
13. Find the HCF and LCM of 8, 48 and 160, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
14. Find the HCF and LCM of 9, 54 and 81, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
15. Find the HCF and LCM of 12, 90 and 150, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
16. Find the HCF and LCM of 15, 60 and 105, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
17. Find the HCF and LCM of 10, 45 and 75, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
18. Find the HCF and LCM of 14, 84 and 196, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
19. Find the HCF and LCM of 16, 64 and 96, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
20. Find the HCF and LCM of 18, 108 and 144, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
21. Find the HCF and LCM of 20, 100 and 140, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]
22. Find the HCF and LCM of 21, 63 and 126, using the prime factorisation method. Show that the product of these three numbers is not equal to the product of their HCF and LCM. [4 Marks]