REAL NUMBERS-KEY POINTS

(A) Main Concepts and Results

• Euclid’s Division Lemma :

Given two positive integers a and b, there exist unique

integers q and r satisfying a = bq + r,  0 £ r < b.

• Euclid’s Division Algorithm to obtain the HCF of two positive integers, say c and d, c > d.

Step 1 : Apply Euclid’s division lemma to c and d, to find whole numbers q and r, such that c = dq + r,  

 0 £r < d.

Step 2 : If r = 0, d is the HCF of c and d. If r ≠¹ 0, apply the division lemma to d and r.

Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

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

• Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

• √2 , √3 , √5 are irrational numbers.

• The sum or difference of a rational and an irrational number is irrational.

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

• For any two positive integers a and b,

    HCF (a, b) × LCM (a, b) = a × b.

• Let x = p/q , p and q are co-prime, be a rational number whose decimal expansion terminates. Then, the prime factorisation of q is of the form 2^m.5ⁿ; m, n are

non-negative integers.

• Let x =  p/q be a rational number such that the prime factorisation of q is not of the form 2^m.5ⁿ; m, n being non-negative integers. Then, x has a non-terminating

repeating decimal expansion.

(B) Multiple Choice Questions

Choose the correct answer from the given four options:

Sample Question 1 : The decimal expansion of the rational number 33/ 2² x 5 will terminate after

(A) one decimal place (B) two decimal places

(C) three decimal places (D) more than 3 decimal places

Solution : Answer (B)

Sample Question 2 : Euclid’s division lemma states that for two positive integers a

and b, there exist unique integers q and r such that a = bq + r, where r must satisfy

(A) 1 < r < b (B) 0 < r £ b

(C) 0 £ r < b (D) 0 < r < b

Solution : Answer (C)

EXERCISE 1.1

Choose the correct answer from the given four options in the following questions:

1. For some integer m, every even integer is of the form

(A) m (B) m + 1

(C) 2m (D) 2m + 1

2. For some integer q, every odd integer is of the form

(A) q (B) q + 1

(C) 2q (D) 2q + 1

3. n² – 1 is divisible by 8, if n is

(A) an integer (B) a natural number

(C) an odd integer (D) an even integer

4. If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is

(A) 4 (B) 2

(C) 1 (D) 3

5. The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is

(A) 13 (B) 65

(C) 875 (D) 1750

6. If two positive integers a and b are written as

a = x³y² and b = xy³; x, y are prime numbers, then HCF (a, b) is

(A) xy (B) xy² (C) x³y³ (D) x²y²

7. If two positive integers p and q can be expressed as

p = ab² and q = a³b; a, b being prime numbers, then LCM (p, q) is

(A) ab (B) a²b² (C) a³b² (D) a³b³

8. The product of a non-zero rational and an irrational number is

(A) always irrational (B) always rational

(C) rational or irrational (D) one

9. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

(A) 10 (B) 100 (C) 504 (D) 2520

10. The decimal expansion of the rational number

14587/1250 will terminate after:

(A) one decimal place (B) two decimal places

(C) three decimal places (D) four decimal places

(C) Short Answer Questions with Reasoning

Sample Question 1: The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify your answer.

Solution : No.

According to Euclid’s division lemma,

a = 3q + r, where 0 £ r < 3 and r is an integer. Therefore, the values of r can be 0, 1 or 2.

Sample Question 2: Can the number 6ⁿ, n being a natural number, end with the digit 5? Give reasons.

Solution : No, because 6n = (2 × 3)ⁿ = 2ⁿ × 3ⁿ, so the only primes in the factorisation of 6n are 2 and 3, and not 5. Hence, it cannot end with the digit 5.

EXERCISE 1.2

1. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

2. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

3. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.

4. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

5. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

6. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.

7. Explain why 3 × 5 × 7 + 7 is a composite number.

8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

9. Without actually performing the long division, find if

987/ 10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

10. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form p/q ? Give reasons.

(D) Short Answer Questions

Sample Question 1: Using Euclid’s division algorithm, find which of the following

pairs of numbers are co-prime:

(i) 231, 396 (ii) 847, 2160

Solution : Let us find the HCF of each pair of numbers.

(i) 396 = 231 × 1 + 165

231 = 165 × 1 + 66

165 = 66 × 2 + 33

66 = 33 × 2 + 0

Therefore, HCF = 33. Hence, numbers are not co-prime.

(ii) 2160 = 847 × 2 + 466

847 = 466 × 1 + 381

466 = 381 × 1 + 85

381 = 85 × 4 + 41

85 = 41 × 2 + 3

41 = 3 × 13 + 2

3 = 2 × 1 + 1

2 = 1 × 2 + 0

Therefore, the HCF = 1. Hence, the numbers are co-prime.

Sample Question 2: Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.

Solution: Any positive odd integer is of the form 2q + 1, where q is a whole number.

Therefore, (2q + 1)² = 4q² + 4q + 1 = 4q (q + 1) + 1, (1)

q (q + 1) is either 0 or even. So, it is 2m, where m is a whole number.

Therefore, (2q + 1)² = 4.2 m + 1 = 8 m + 1. [From (1)]

Sample Question 3: Prove that √2 + √3 is irrational.

Solution : Let us suppose that 2 + √3 is rational. Let 2 +√ 3 = a , where a is rational.

Therefore, √2 = a- √3

Squaring on both sides, we get

2 = a² + 3 – 2a √3

Therefor

√ 3= a² +1/2a

 which is a contradiction as the right hand side is a rational number while√ 3 is irrational. Hence, √2 + √3 is irrational.

EXERCISE 1.3

1. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

2. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

3. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

4. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

5. Show that the square of any odd integer is of the form 4q + 1, for some integer q.

6. If n is an odd integer, then show that n2 – 1 is divisible by 8.

7. Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.

8. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

9. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

10. Prove that 3+ 5 is irrational.

11. Show that 12n cannot end with the digit 0 or 5 for any natural number n.

12. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

13. Write the denominator of the rational number

257/ 5000 in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

14. Prove that p + q is irrational, where p, q are primes.

(E) Long Answer Questions

Sample Question 1 : Show that the square of an odd positive integer can be of the

form 6q + 1 or 6q + 3 for some integer q.

Solution : We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2,

6m + 3, 6m + 4 or 6m + 5, for some integer m.

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5

Thus we have:

(6 m +1)² = 36 m² + 12 m + 1 = 6 (6 m² + 2 m) + 1 = 6 q + 1, q is an integer

(6 m + 3)² = 36 m² + 36 m + 9 = 6 (6 m² + 6 m + 1) + 3 = 6 q + 3, q is an integer

(6 m + 5)² = 36 m² + 60 m + 25 = 6 (6 m² + 10 m + 4) + 1 = 6 q + 1, q is an integer.

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.

EXERCISE 1.4

1. Show that the cube of a positive integer of the form 6q + r, q is an integer and

r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

2. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

3. Prove that one of any three consecutive positive integers must be divisible by 3.

4. For any positive integer n, prove that n3 – n is divisible by 6.

5. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q+1, 5q+2, 5q+3, 5q+4].

Question-Bank

Section-A  Objectives(1 mark Q.)

1)   Express 140 as a product of its prime factors.

2)   Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3)   Find the LCM and HCF of 6 and 20 by the prime factorization method.

4)   State whether  3125/13 will have a terminating decimal expansion or a non-terminating repeating decimal.

5)   State whether 8/17 will have a terminating decimal expansion or a non-terminating  repeating decimal.

6)   Express 156 as a product of its prime factors.

7)   Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.

8)   Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.

9)   State whether  15/6 will have a terminating decimal expansion or a non-terminating repeating decimal.

10)                      State whether 50/35 will have a terminating decimal expansion or a non-terminating    repeating decimal.

11)                     Express 3825 as a product of its prime factors.

12)                     Find the LCM and HCF of 8, 9 and 25 by the prime factorization method.

13)                     Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.

14)                     State whether 343/29 will have a terminating decimal expansion or a non-terminating repeating decimal.

¹⁵⁾   State whether  23/ 2³ X 5² will have a terminating decimal expansion or a non-terminating repeating decimal.

16)                     Express 5005 as a product of its prime factors.

17)                     Find the LCM and HCF of 24, 36 and 72 by the prime factorization method.

18)                     Find the LCM and HCF of 96 and 404 by the prime factorization method.

19)                     State whether 455/64 will have a terminating decimal expansion or a non-terminating repeating decimal.

20)                     State whether 1600/15 will have a terminating decimal expansion or a non-terminating repeating decimal.

21)                     What is the maximum no. of factors of a prime number?

22)                     Given HCF of (16, 100) = 4. Find L.C.M of L (16, 100).

23)                     Write a rational no. between √2 and √3.

24)                     Write if 343/28 is a terminating or non-terminating repeating decimal without doing actual division.

25)                     Tell whether the prime factorization of 15 is 1× 3× 5 or not.

26)                     If x and y are two irrational numbers then tell whether  x - y is always irrational or not.

27)                      What is the L.C.M of x and y is a multiple of x?

28)                      Write the sum of exponents of prime factors of 98.

29)                      State if (√2 – √3) (√2 + √3) is rational or irrational.

30)                     Express 0.03 as a rational number in the form of p/q.

Section-B(3 marks Q.)

31)                      Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

32)                     Use Euclid’s division algorithm to find the HCF of 135 and   225

33)                     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.

34)                     Prove that 3 is irrational.

35)                      Show that 5 – root 3 is irrational.

36)                      Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

37)                      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?

38)                      Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.

39)                      Use Euclid’s algorithm to find the HCF of 4052 and 12576.

40)                     Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.

41)                      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.

42)                      Prove that 3- root 2 5 is irrational.

43)                      Prove that Root 2 is irrational.

44)                     In a school there are two sections- section A and Section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.

45)                      Find the LCM and HCF of 336 and 54 and verify that LCM × HCF = product of the two numbers.

46)                     Use Euclid’s division algorithm to find the HCF of 867 and 255

47)                      Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

48)                     Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,9m + 1 or 9m + 8

49)                     Prove that 7 5 is irrational.

50)                      Prove that 5 is irrational.

51)                      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?

52)                      Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two numbers.

53)                     Use Euclid’s division algorithm to find the HCF of 196 and 38220

54)                      Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,9m + 1 or 9m + 8.

55)                     Show that every positive odd integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

56)                      Show that√ 3+√ 2 is irrational.

57)                      Prove that 3 + √ 5 is irrational.

58)                      A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of barfis that can be placed in each stack for this purpose?

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

60)                      Show that one and only one out of n, n+4, n+8, n+12 and n+ 16 is division by 5 where n is any positive integer.

61)                      Show that the sum and product of two irrational numbers 7+√5 and 7-√5 are rational numbers

62)                     Use Euclid’s division lemma to find the H.C.F of 615 and 154.