BOARD EXAM QUESTIONS
PRIME FACTORISATION, LCM AND HCF
MULTIPLE CHOICE QUESTIONS
1. The prime factorisation of natural number 288 is
(a) $2^4 \times 3^3$
(b) $2^4 \times 3^2$
(c) $2^5 \times 3^2$
(d) $2^5 \times 3^1$
2. The prime factorisation of 432 is
(a) $2^3 \times 3^4$
(b) $2^4 \times 3^3$
(c) $2^3 \times 3^3$
(d) $2^4 \times 3^4$
3. The prime factorisation of 1728 is
(a) $2^5 \times 3^3$
(b) $2^5 \times 3^4$
(c) $2^6 \times 3^3$
(d) $2^6 \times 3^2$
4. If the HCF of 360 and 64 is 8, then their LCM is
(a) 2480
(b) 2780
(c) 512
(d) 2880
5. If the HCF of 72 and 234 is 18, then the LCM(72,234) is
(a) 936
(b) 836
(c) 324
(d) 234
6. (HCF $\times$ LCM) for the numbers 70 and 40 is
(a) 10
(b) 280
(c) 2800
(d) 70
7. (HCF $\times$ LCM) for the numbers 30 and 70 is
(a) 2100
(b) 21
(c) 210
(d) 70
8. LCM of $(2^3 \times 3 \times 5)$ and $(2^4 \times 5 \times 7)$ is
(a) 40
(b) 560
(c) 1680
(d) 1120
9. If HCF(72, 120)=24, then LCM(72, 120) is
(a) 72
(b) 120
(c) 360
(d) 9640
10. The prime factorisation of the number 2304 is
(a) $2^8 \times 3^2$
(b) $2^7 \times 3^3$
(c) $2^8 \times 3^1$
(d) $2^7 \times 3^2$
11. If n is a natural number, then $8^n$ cannot end with digit
(a) 0
(b) 2
(c) 4
(d) 6
12. The HCF of the smallest 2-digit number and the smallest composite number is
(a) 4
(b) 20
(c) 2
(d) 10
13. The prime factorisation of the number 5488 is
(a) $2^3 \times 7^3$
(b) $2^4 \times 7^3$
(c) $2^4 \times 7^4$
(d) $2^3 \times 7^4$
14. If two positive integers x and y are written as $x=a^3b^2$ and $y=ab^3$, where a and b are prime numbers, then their HCF(x,y) is
(a) $ab$
(b) $ab^2$
(c) $a^3b^3$
(d) $a^2b^2$
15. The product of the HCF and LCM of two numbers 50 and 20 is
(a) 10
(b) 100
(c) 1000
(d) 20
16. If 'p' and 'q' are natural numbers and 'p' is the multiple of 'q', then what is the HCF of 'p' and 'q'?
(a) pq
(b) p
(c) q
(d) p+q
17. If 'n' is a natural number, then which of the following numbers end with zero?
(a) $(3 \times 2)^n$
(b) $(2 \times 5)^n$
(c) $(6 \times 2)^n$
(d) $(5 \times 3)^n$
18. The ratio of HCF to LCM of the least composite number and the least prime number is
(a) 1:2
(b) 2:1
(c) 1:1
(d) 1:3
19. The LCM of smallest 2-digit number and smallest composite number is
(a) 12
(b) 4
(c) 20
(d) 40
20. The greatest number which divides both 30 and 80 leaving remainders 2 and 3 respectively
(a) 10
(b) 7
(c) 11
(d) 14
21. The ratio of HCF and LCM of the least prime number and the least composite number is
(a) 1:2
(b) 2:1
(c) 1:3
(d) 1:1
22. The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, the other number is
(a) 36
(b) 45
(c) 9
(d) 81
23. If a and b are two consecutive natural numbers, then HCF(a,b) is
(a) a
(b) b
(c) ab
(d) 1
24. The greatest number which divides both 83 and 138, leaving remainders 5 and 8 respectively is
(a) 13
(b) 65
(c) 26
(d) 39
25. HCF of 144 and 198 is
(a) 9
(b) 18
(c) 6
(d) 12
26. 225 can be expressed as
(a) $5 \times 3^2$
(b) $5^2 \times 3$
(c) $5^2 \times 3^2$
(d) $5^3 \times 3$
27. What is the largest number that divides 245 and 1029 leaving remainder 5 in each?
(a) 15
(b) 16
(c) 9
(d) 5
28. The simplest form of $\frac{1095}{1168}$ is
(a) $\frac{17}{26}$
(b) $\frac{25}{26}$
(c) $\frac{13}{16}$
(d) $\frac{15}{16}$
29. Given that HCF(156, 78)=78, LCM(
156, 78) is
(a) 156
(b) 78
(c) 156 $\times$ 78
(d) 156 $\times$ 2
30. 120 can be expressed as a product of its prime factors as
(a) $5 \times 8 \times 3$
(b) $15 \times 2^3$
(c) $10 \times 2^2 \times 3$
(d) $5 \times 2^3 \times 3$
31. 180 can be expressed as a product of its prime factors as
(a) $10 \times 2 \times 3^2$
(b) $25 \times 4 \times 3$
(c) $2^2 \times 3^2 \times 5$
(d) $4 \times 9 \times 5$
32. The HCF of 135 and 225 is
(a) 9
(b) 15
(c) 45
(d) 25
33. 840 can be expressed as a product of prime numbers are
(a) $2^6 \times 6 \times 5 \times 7$
(b) $2^3 \times 3 \times 5 \times 7$
(c) $2 \times 3 \times 4 \times 5 \times 7$
(d) $3 \times 5 \times 7 \times 8$
34. The total number of factors of a prime number is
(a) 1
(b) 0
(c) 2
(d) 3
35. The HCF and the LCM of 12, 21, 15 respectively are
(a) 3, 140
(b) 12, 420
(c) 3, 420
(d) 420, 3
36. The sum of exponents of prime factors in the prime factorisation of 196 is
(a) 3
(b) 4
(c) 5
(d) 2
37. The HCF of 135 and 225 is
(a) 15
(b) 75
(c) 45
(d) 5
38. The exponent of 2 in the prime factorisation of 144 is
(a) 2
(b) 4
(c) 1
(d) 6
39. The LCM of the smallest 2 digit number and the largest multiple of 6 which is less than 50 is
(a) 2
(b) 48
(c) 120
(d) 240
40. The HCF of 112 and 196 is
(a) 7
(b) 14
(c) 28
(d) 56
41. 1260 can be expressed as a product of prime numbers as
(a) $3^2 \times 4 \times 5 \times 4$
(b) $3^2 \times 2^2 \times 5 \times 7^2$
(c) $3^2 \times 2^2 \times 5 \times 7$
(d) $2^2 \times 3^2 \times 5^2 \times 7$
42. The HCF of the smallest number of two digits and the largest multiple of 5 which is less than 40 is
(a) 3
(b) 4
(c) 5
(d) 10
VERY SHORT ANSWERS / SHORT ANSWERS
1. Find the LCM and HCF of 92 and 510 using prime factorisation.
2. Find the HCF of the number 540 and 630 using prime factorisation method.
3. Show that $(15)^n$ cannot end with the digit 0 for any natural number n.
4. Find LCM of 576 and 512 by prime factorisation.
5. Find HCF of 660 and 704 by prime factorisation.
6. Find LCM of 480 and 256 using prime factorisation.
7. Find the greatest number which divides 85 and 72 leaving remainder 1 and 2 respectively.
8. Find the least number which when divided by 12, 16 and 24 leaves remainder 7 in each case.
9. Find by prime factorisation, the LCM of the numbers 18180 and 7575. Also find the HCF of the two numbers.
10. Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells ring at 6 a.m., when will they ring together again?
11. Prove that $4^n$ can never end with digit 0, where n is a natural number.
12. Two numbers are in the ratio 2:3 and their LCM is 180. What is the HCF of these numbers?
13. Using prime factorisation, find the HCF and LCM of 96 and 120.
14. 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?
15. Find the greatest 3 digit number which is divisible by 18, 24 and 36.
16. Show that $6^n$ cannot end with digit 0 for any natural number n.
17. Find HCF and LCM of 72 and 120.
18. Find the HCF and LCM of 26, 65 and 117 using prime factorisation.
19. Show that $8^n$ can never end with digit 0 for any natural number n.
20. Find LCM and HCF of 96 and 160 using prime factorisation method.
21. Show that $12^n$ can never end with a digit 0, where n is a natural number.
(or)
Check whether $12^n$ can end with the digit 0 for any natural number n.
22. Find the HCF and the LCM of 36 and 60, using prime factorisation method.
23. Check whether $6^n$ can end with the digit 0, for any natural number n.
24. Find the HCF and LCM of 84, 90 and 120 by prime factorisation method.
25. Express 288 as product of its prime factors.
26. Find the LCM and HCF of two numbers 26 and 91 by the method of prime factorisation.
27. For two numbers x and y, if xy=1344 and HCF(x,y)=8, then find LCM(x,y).
28. Find the HCF of 96 and 404 by prime factorisation.
29. Express 792 as the product of its prime factors.
30. Find the exponent of 2 in the prime factorisation of 288.
31. Explain why $2 \times 3 \times 5+5$ and $5 \times 7 \times 11+7 \times 5$ are composite numbers.
32. State and give reason whether $5 \times 7 \times 11+11$ is a composite number or prime number.
33. If the sum of LCM and HCF of two numbers is 1260 and the LCM is 900 more than their HCF, find their LCM.
34. An army contingent of 612 members is to march behind an army band of 48 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?
35. Find HCF of 44, 96 and 404 by prime factorisation method. Hence find their LCM.
36. Given that the HCF of two numbers is 11 and their LCM is 693. If one of the numbers is 77, then find the other number.
37. Check whether $6^n$ can end with the digit 0 (zero) for any natural number n.
38. Find the LCM of 150 and 200.
39. Show that $12^n$ cannot end with digit 0 or 5 for any natural number n.
40. Given that HCF(135, 225) =40, find the LCM(135, 225).
41. Given that HCF(120, 160)=40, find LCM(120, 160).
42. The LCM of two numbers is 182 and their HCF is 13. If one of the numbers is 26, find the other.
43. Find the HCF of 12, 18 and 30.
44. If the HCF of 65 and 117 is written as 65m-117, then find the value of m.
45. Find the largest positive integer that will divide 122, 150 and 115 leaving remainders 5, 7 and 11 respectively.
46. It is given that HCF(504, 2200)=8, then find LCM(504,2200).
47. Define a prime number and a composite number. Hence explain why $7 \times 11 \times 13+13$ is a composite number.
48. Check whether $9^n$ can end with the digit 0 for any positive integer n.
49. Write the HCF of $x^3y^4z^2$ and $x^2y^3z^5$ where x, y, z are distinct prime numbers.
50. The HCF of 119 and 175 is expressed as $18m-83$. Find the value of m.
51. Is $3 \times 5 \times 7 \times 11$ a composite number? Give reason for your answer.
52. The LCM of two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find the HCF of the two numbers.
53. The HCF of two numbers is 116 and their LCM is 1740. If one number is 580, find the other.
54. Two positive integers a and b can be written as $a=x^3y^2$ and $b=xy^3$, x, y are prime numbers. Find LCM(a,b).
55. If HCF(336, 54)=6, find LCM(336,54).
56. Write the smallest number which is divisible by both 306 and 657.
57. Express 429 as a product of its prime factors.
58. If HCF of 65 and 117 is expressible in the form $65n-117$, then find the value of n.
59. On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
60. Find the largest number which on dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively.
61. Write the number of zeroes in the end of a number whose prime factorisation is $2^2 \times 5^3 \times 3^2 \times 17$.
62. The HCF of two numbers a and b is 5, and their LCM is 200. Find the product ab.
63. Find the HCF of 612 and 1314 using prime factorisation.
IRRATIONAL NUMBERS
MULTIPLE CHOICE QUESTIONS
1. The number $(5-3\sqrt{5}+\sqrt{5})$ is
(a) an integer
(b) a rational number
(c) an irrational number
(d) a whole number
2. The product of a non-zero rational number and an irrational number is
(a) always irrational
(b) always rational
(c) rational or irrational
(d) always positive
3. If $p^2=\frac{32}{50}$, then p is a/an
(a) whole number
(b) integer
(c) rational number
(d) irrational number
4. Which of the following is an irrational number?
(a) $(2\sqrt{3}-\frac{1}{\sqrt{3}})^2$
(b) $(\sqrt{2}-1)^2$
(c) $\sqrt{2}-(2+\sqrt{2})$
(d) $\frac{\sqrt{2}+5\sqrt{2}}{\sqrt{2}}$
5. $2\sqrt{3}$ is
(a) an integer
(b) a rational number
(c) an irrational number
(d) a whole number
6. Which of the following is the decimal expansion of an irrational number?
(a) 3.14
(b) 3.3333...
(c) 6.010010001...
(d) 7.25
FILL IN THE BLANKS
$\frac{2+\sqrt{5}}{3}$ is ____________ number.
VERY SHORT ANSWERS / SHORT ANSWERS
1. Prove that $\sqrt{5}$ is an irrational number.
2. Prove that $\sqrt{3}$ is an irrational number.
3. Prove that $\sqrt{2}$ is an irrational number.
4. Prove that $\sqrt{2}+\sqrt{5}$ is irrational.
5. Find a rational number between $\sqrt{2}$ and $\sqrt{3}$.
6. Find a rational number between $\sqrt{2}$ and $\sqrt{7}$.
7. Write one irrational number between 0.15 and 0.21.
8. Write any one irrational number between 1 and 2.
9. Prove that $3+7\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
10. Prove that $5-\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
11. Prove that $4+2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
12. Prove that $2-3\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.
13. Prove that $10+2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
14. Prove that $7-2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
15. Prove that $8+5\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.
16. Prove that $11+3\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
17. Prove that $3-2\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.
18. Prove that $5+2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
19. Prove that $7+4\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.
20. Prove that $2+\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
21. Prove that $6-\sqrt{7}$ is an irrational number, given that $\sqrt{7}$ is an irrational number.
22. Prove that $2+3\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.
23. Prove that $3+2\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
24. If $\sqrt{2}$ is given as an irrational number, then prove that $5-2\sqrt{2}$ is an irrational number.
25. Prove that $7\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
26. Prove that $3+\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
27. Given that $\sqrt{2}$ is irrational, prove that $3\sqrt{2}$ is also irrational.
28. Given that $\sqrt{3}$ is an irrational number, prove that $5+2\sqrt{3}$ is an irrational number.
29. If $\sqrt{2}$ is given as an irrational number, then prove that $7-2\sqrt{2}$ is an irrational number.
30. Given that $\sqrt{3}$ is irrational, prove that $5\sqrt{3}-2$ is an irrational number.
31. Show that $5+2\sqrt{7}$ is an irrational number, where $\sqrt{7}$ is given to be an irrational number.
32. Assuming that $\sqrt{3}$ is an irrational number, prove that $5\sqrt{3}-7$ is an irrational number.
33. Taking $\sqrt{5}$ to be an irrational number, prove that $3\sqrt{5}-7$ is also an irrational number.
34. Prove that $2+5\sqrt{3}$ is an irrational number, it is being given that $\sqrt{3}$ is an irrational number.
35. Prove that $4-5\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
36. Prove that $3\sqrt{3}-7$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
37. Given that $\sqrt{3}$ is an irrational number, prove that $4\sqrt{3}-7$ is also an irrational number.
38. Show that $\frac{3+\sqrt{7}}{5}$ is an irrational number, given that $\sqrt{7}$ is an irrational number.
39. Prove that $\frac{2+\sqrt{3}}{5}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
40. Prove that $2+5\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
41. Prove that $2+3\sqrt{3}$ is an irrational number, when it is given that $\sqrt{3}$ is an irrational number.
42. Show that $\frac{2+3\sqrt{2}}{7}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
43. Prove that $5-3\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
44. Give example of two irrational numbers,whose
(i) sum is a rational number, (ii) product is an irrational number.