__Examples__
1. Find the H.C.F. of 2

^{3}X 3^{2}X 5 X 7^{4}, 2^{2}X 3^{5}X 5^{2}X 7^{3}, 2^{3}X5^{3}X7^{2}**2.**

**Find the H.C.F. of 108, 288 and 360.**

3. Find the H.C.F. of 513, 1134 and 1215

4. Find the L.C.M. of 2

^{2}x 3^{3}x 5 x 7^{2}, 2^{3}x 3^{2}x 5^{2}x 7^{4}, 2 x 3 x 5^{3}x 7 x 11.
5. Find the L.C.M. of 72, 108 and 2100

6. Find the L.C.M. of 16, 24, 36 and 54

7. Find the HCF and LCM of 2/3 , 8/9, 16/81, and 10/27

8. Find the HCF and LCM of 0.63, 1.05 and 2.1

9. Two numbers are in the ratio of 15:11. If their HCF is 13, find the numbers

10. The HCF of two numbers is 11 and their LCM is 693. If one of the numbers is 77, find the other

11. Find the greatest possible length which can be used to measure exactly the lengths 4m 95 cm, 9m and 16m 65cm

12. Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.

13. Find the largest number which divides 62, 132 and 237 to leave the same remainder in each case

14. Find the least number exactly divisible by 12, 15, 20 and 27

15. Find the least number which when divided by 6, 7, 8, 9 and 12 leaves the same remainder 1 in each case

16. Find the largest number of four digits exactly divisible by 12, 15, 18 and 27

17. Find the smallest number of five digits exactly divisible by 16, 24, 36 and 54

18. Find the least number which when divided by 20, 25, 35 and 40 leaves remainders 14, 19, 29 and 34 respectively

19. Find the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3 but when divided by 9 leaves no remainder.

20. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively .If they all change simultaneously at 8:20:00 hours, then at what time they again change simultaneously

21. Arrange the fractions 17/18, 31/36, 43/45, 59/60 in the ascending order.

__Practice Questions__
1. Which of the following has most number of divisors? A. 99 B. 101 C. 176 D. 182

2. The H.C.F. of 2

^{2}* 3^{3}* 5^{5}, 2^{3}* 3^{2}* 5^{2}* 7 and 2^{4}* 3^{4}* 5 * 7^{2}* 11 is :
3. H.C.F. of 4 * 27 * 3125, 8 * 9 * 25 * 7 & 16 * 81 * 5 * 11 * 49 is :

4. Which of the following is a pair of co - primes? A. (16, 62) B. (18, 25) C. (21, 35) D. (23, 92)

5. The L.C.M. of 2

^{3}* 3^{2}* 5 * 11, 2^{4}* 3^{4}* 5^{2}* 7 and 2^{5}* 3^{3}* 5^{3}* 7^{2}* 11 is :
6. Find the HCF and LCM of 2/3, 3/5, 4/7, 9/13 is

7. The G.C.D. of 1.08, 0.36 and 0.9 is :

8. H.C.F. of 3240, 3600 and a third number is 36 and their L.C.M. is 2

^{4}* 3^{5}* 5^{2}* 7^{2}. The 3^{rd}no is:
9. The L.C.M. of 3, 2.7 and 0.09 is :

10. The ratio of two numbers is 3: 4 and their H.C.F. is 4. Their L.C.M. is :

11. The sum of two numbers is 216 and their H.C,F. is 27. The numbers are :

12. Three numbers are in the ratio 1 : 2 : 3 and their H.C.F. is 12. The numbers are :

13. The sum of two numbers is 528 and their H.C.F. is 33. The number of pairs of numbers satisfying the above conditions is :

14. The number of number - pairs lying between 40 and 100 with their H.C.F. as 15 is :

15. The H.C.F. of two numbers is 12 and their difference is 12. The numbers are :

16. The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is :

17. The product of two numbers is 2028 and their H.C.F. is 13. The number of such pairs is :

18. Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is :

19. The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. The sum of the No’s

20. The sum of two numbers is 2000 and their L.C.M. is 21879. The two numbers are :

21. Three numbers are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is :

22. The H.C.F. of two numbers is 11 and their L.C.M. is 7700. If one of the numbers is 275, then the other is :

23. The H.C.F. and L.C.M. of two numbers are 84 and 21 respectively. If the ratio of the two numbers is 1 : 4, then the larger of the two numbers is :

24. The L.C.M. of two numbers is 495 and their H.C.F. is 5. If the sum of the numbers is 100, then their difference is :

25. The product of the L.C.M. and H.C.F. of two numbers is 24. The difference of two numbers is 2. Find the numbers.

26. The L.C.M. of two numbers is 45 times their H.C.F. If one of the numbers is 125 and the sum of H.C.F. and L.C.M. is 1150, the other number is :

27. Product of two co - prime numbers is 117. Their L.C.M. should be :

28. The H.C.F. and L.C.M. of two numbers are 50 and 250 respectively. If the first number is divided by 2, the quotient is 50. The second number is :

29. The L.C.M. of three different numbers is 120. Which of the following cannot be their H.C.F?

A. 8 B. 12 C. 24 D. 35

30. If the sum of two numbers is 55 and the H.C.F. and L.C.M. of these numbers are 5 and 120 respectively, then the sum of the reciprocals of the numbers is equal to :

31. The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is :

32. About the number of pairs which have 16 as their H.C.F. and 136 as their L.C.M., we can definitely say that :

33. The H.C.F. of two numbers is 8. Which one of the following can never be their L.C.M?

A. 24 B. 48 C. 56 D. 60

34. L.C.M. of two prime numbers x and y (x > y) is 161. The value of 3y - x is :

35. Two numbers, both greater than 29, have H.C.F. 29 and L.C.M. 4147. The sum of the No’s is

36. The H.C.F. and L.C.M. of two numbers are 11 and 385 respectively. If one number lies between 75 and 125, then that number is :

37. The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is :

38. Three different containers contain 496 litres, 403 litres and 713 litres of mixtures of milk and water respectively. What biggest measure can measure all the different quantities exactly?

39. A rectangular courtyard 3.78 metres long and 5.25 metres wide is to be paved exactly with square tiles, all of the same size. What is the largest size of the tile which could be used for the purpose?

40. The maximum number of students among them 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is :

41. Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

42. Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is :

43. The greatest number which can divide 1356, 1868 and 2764 leaving the same remainder 12 in each case, is :

44. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 is

45. What will be the least number which when doubled will be exactly divisible by 12, 18, 21 & 30

46. The least number of five digits which is exactly divisible by 12, 15 and 18, is .

47. The least number which is a perfect square and is divisible by each of the numbers 16, 20 &24:

48. The greatest number of four digits which is divisible by 15, 25, 40 and 75 is :

49. The smallest number which when diminished by 7, is divisible by 12, 16, 18, 21 and 28 is :

50. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is :

51. The least number which when increased by 5 is divisible by each one of 24, 32, 36 and 54, is

52. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is :

53. The largest four-digit number which when divided by 4, 7 or 13 leaves a remainder of 3 in each case, is :

54. Let the least number of six digits, which when divided by 4, 6, 10 and 15, leaves in each case the same remainder of 2, be N. The sum of the digits in N is :

55. The least number which when increased by 5 is divisible by each one of 24, 32, 36 and 54, is

56. The least number, which when divided by 48, 60, 72, 108 and 140 leaves 38, 50, 62, 98 and 130 as remainders respectively, is :

57. Find the least multiple of 23, which when divided by 18, 21 and 24 leaves remainders 7, 10 and 13 respectively.

58. A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. After what time will they meet again at the starting point?

59. Four different electronic devices make a beep after every 30 minutes, 1 hour, 1 ½ hour and 1 hour 45 minutes respectively. All the devices beeped together at 12 noon. They will again beep together at :

60. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?

61. Find the least number which when divided by 16, 18, 20 and 25 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder,

62. The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is :