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 whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

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

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

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. 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

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

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

12. 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?

13. Express 156 as a product of its prime factors.

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

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

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

17. State whether35/50 will have a terminating decimal expansion or a non-terminating repeatingdecimal.

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

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

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

21. 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.

22. Prove that 3√2 5 is irrational.

23. Prove that 1/√2 is irrational. (3 marks)

24. In a school there are tow 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.

25. Express 3825 as a product of its prime factors.

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

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

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

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

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

31. Use Euclid’s division algorithm to find the HCF of 867 and 255

32. 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.

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

34. Prove that 7 √5 is irrational.

35. Prove that √5 is irrational.

36. 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?

37. Express 5005 as a product of its prime factors.

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

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

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

41. State whether15/ 1600 will have a terminating decimal expansion or a non-terminating repeating decimal.

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

43. Use Euclid’s division algorithm to find the HCF of 196 and 38220 (3 marks)

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

45. 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

46. Show that 3√ 2 is irrational.

47. Prove that 3 + 2 √5 is irrational.

48. A sweet seller 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?

49. Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

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

51. 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? Sol. Hints: Find the HCF of 616 and 32

52. 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.]

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

54. Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

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

56. Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM.

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

58. Find the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y

59. Prove that no number of the type 4K + 2 can be a perfect square.

60. Express each number as a product of its prime factors:(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

61. 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

62. Find the LCM and HCF of the following integers by applying the prime factorization method. 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

63. Given that HCF (306, 657) = 9, find LCM (306, 657).

64. Check whether 6n can end with the digit 0 for any natural number n.

65. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

66. 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?

67. Use Euclid’s division lemma to show that the square of any positive integer is of the form 5q, 5q+1,5q+4 for some integer q.

68. Show that any one of the numbers (n + 2), n and (n + 4) is divisible by 3.

69. If 793800 = 2 3 x 3 m x 5 n x 7 2, find the value of m and n.

70. If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y.

71. Given that HCF (135, 225) = 45. Find LCM (135, 225).

72. Solve √18x √50. What type of number is it, rational or irrational?

73. What type of decimal expansion will 69/60 represent? After how many places will the decimal expansion terminate?

74. Find the H.C.F. of the smallest composite number and the smallest prime number.

75. If a = 4q + r then what are the conditions for a and q. What are the values that r can take?

76. What is the smallest number by which √5 - √ 3 be multiplied to make it a rational no? Also find the no. so obtained.

77. What is the digit at unit’s place of 96?

78. Find one rational and one irrational no. between √3 and √5.

79. If the no. p never to end with the digit 0 then what are the possible value (s) of p?

80. State Euclid’s Division Lemma and hence find HCF of 16 and 28.

81. State fundamental theorem of Arithmetic and hence find the unique fraternization of 120.

82. Prove that 1/(2 - √5) is irrational number.

83. Check whether 5 × 7 × 11 + 6 is a composite number.

84. Check whether 7 × 6 × 3 × 5 + 5 is a composite number.

85. Find all positive integral values of n for which n2+96 is perfect square

Topics:

1. Euclid's Division Lemma/Algorithm

2. Fundamental Theorem of Arithmetic

3. Irrational Numbers

4. Decimal expression of Rational Number

## 6 comments:

Q. Prove that no number of the type 4k+2 can be a perfect square.

Ans: If p is a prime factor of a perfect square, p2 must also be a factor of that perfect square. 4k+2 = 2(2k+1)

2 is a factor of 4k+2 but 2k+1 is odd and cannot have factor 2, so 4k+2 is not divisible by 4, and therefore cannot be a perfect square.

Q. 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?

Ans: We need to calculate the LCM to find the answer. 18 = 2x3x3 ;12= 2x2x3 ; LCM = 36

Q. show that only one out of n, n+2 or n+4 is divisible by 3 where n is positive integer.

Solution: When n = 1, exactly one of 1, 1+2, and 1+4 is divisible by 3,

namely 1+2, since 3 is divisible by 3, and the other two 1 and 5 are not.

Suppose for n < k, only one out of n, n+2, n+4 is divisible by 3

For n = k, we consider k, k+2, k+4.

By the induction hypothesis, only one of k-1, k+1 and k+3 is

divisible by 3. We look at the three possible cases.

Case 1: k-1 is the one which is divisible by 3. Then k-1 = 3m, for some positive integer m.

Then add 1 to both sides of k-1 = 3m:

k-1+1=3m+1

k = 3m+1 then k is NOT divisible by 3

Now add 3 to both sides of k-1 = 3m:

k-1+3=3m+3

k+2 = 3m+3

k+2 = 3(m+1)

then k+2 IS divisible by 3

Now add 5 to both sides of k-1 = 3m:

k-1+5=3m+5

k+4 = 3m+5

k+4 = 3m+3 + 2 = 3(m+1)+2

then k+4 is NOT divisible by 3.

So, we have proved case 1 for n = k

Case 2: k+1 is the one which is divisible by 3. Then k+1 = 3m, for some positive integer m.

Then add -1 to both sides of k+1 = 3m:

k+1-1=3m-1

k = 3m-1

then k is NOT divisible by 3

Now add 1 to both sides of k+1 = 3m:

k+1+1=3m+1

k+2 = 3m+1

then k+2 is NOT divisible by 3

Now add 3 to both sides of k+1 = 3m:

k+1+3=3m+3

k+4 = 3m+3

k+4 = 3(m+1)

so k+4 IS divisible by 3.

So, we have proved case 2.

Case 3: k+3 is the one which is divisible by 3. Then k+3 = 3m, for some positive integer m.

Then add -3 to both sides of k+3 = 3m:

k+3-3=3m-3

k = 3(m-1) then k IS divisible by 3

Now add -1 to both sides of k+3 = 3m:

k+3-1=3m-1

k+2 = 3m-1 then k+2 is NOT divisible by 3

Now add 1 to both sides of k+3 = 3m:

k+3+1=3m+1

k+4 = 3m+1 then k+4 is NOT divisible by 3.

So, we have proved case 3.

If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y

Let us first find the HCF of 210 and 55.

Applying Euclid division lemna on 210 and 55, we get

210 = 55 × 3 + 45 ....(1)

Since the remainder 45 ≠ 0. So, again applying the Euclid division lemna on 55 and 45, we get

55 = 45 × 1 + 10 .... (2)

Again, considering the divisor 45 and remainder 10 and applying division lemna, we get

45 = 4 × 10 + 5 .... (3)

We now, consider the divisor 10 and remainder 5 and applying division lemna to get

10 = 5 × 2 + 0 .... (4)

We observe that the remainder at this stage is zero. So, the last divisor i.e., 5 is the HCF of 210 and 55.

∴ 5 = 210 × 5 + 55y

⇒ 55y = 5 - 1050 = -1045

⇒ y = -19

Q. finds the H.C.F. of 65 and 117 and express it in the form of 65m+117n.

Answer: Among 65 and 117; 2117 > 65

Since 117 > 65, we apply the division lemma to 117 and 65 to obtain

117 = 65 × 1 + 52 … Step 1

Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain

65 = 52 × 1 + 13 … Step 2

Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain

52 = 4 × 13 + 0 … Step 3

In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers

The H.C.F. of 65 and 117 is 13

From Step 2:

13 = 65 – 52× 1 … Step 4

From Step 1:

52 = 117 – 65 × 1

Thus, from Step 4, it is obtained

13 = 65 – (117 – 65 × 1)

⇒13 = 65 × 2 – 117

⇒13 = 65 × 2 + 117 × (–1)

In the above relationship the H.C.F. of 65 and 117 is of the form 65m + 117 n, where m = 2 and n = –1

Find all positive integral values of n for which n2+96 is perfect square.

Answer: Let n2 + 96 = x2

⇒ x2 – n2 = 96

⇒ (x – n) (x + n) = 96

⇒ both x and n must be odd or both even

on these condition the cases are

x – n = 2, x + n = 48

x – n = 4, x + n = 24

x – n = 6, x + n = 16

x – n = 8, x + n = 12

and the solution of these equations can be given as

x = 25, n = 23

x = 14, n = 10

x = 11, n = 5

x = 10, n = 2

So, the required values of n are 23, 10, 5, and 2.

Q. Prove that one of every three consecutive integers is divisible by 3.

Ans: n,n+1,n+2 be three consecutive positive integers We know that n is of the form 3q, 3q +1, 3q + 2

So we have the following cases

Case – I when n = 3q

In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3

Case - II When n = 3q + 1

put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3

Case – III When n = 3q +2

put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3.

But n and n+1 are not divisible by 3

Hence one of n, n + 1 and n + 2 is divisible by 3

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