Mathematics Assignment: Real Numbers

Topic: 100 Statement Problems on HCF & LCM

Level: Class 10 CBSE (Standard/Basic)

1. 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?
   (Hint: Splitting into equal columns. Calculate the HCF of 616 and 32.)
2. 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?
   (Hint: Least area implies the tallest possible equal stacks. Calculate the HCF of 420 and 130.)
3. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in an exact number of times.
   (Hint: You need a common divisor for both volumes. Calculate the HCF of 850 and 680.)
4. A merchant has three pieces of timber of lengths 42m, 49m, and 63m respectively. He wants to divide them into planks of equal length. What is the greatest possible length of each plank?
   (Hint: Breaking lengths into equal common parts. Calculate the HCF of 42, 49, and 63.)
5. A library has 96 copies of English books and 120 copies of Hindi books. These books are to be stacked in such a way that each stack has the same number of books and the stacks are of the same subject. What is the maximum number of books in each stack?
   (Hint: Splitting different totals into equal groups. Calculate the HCF of 96 and 120.)
6. A fruit vendor has 990 apples and 945 oranges. He packs them into baskets such that each basket contains only one type of fruit and the number of fruits in each basket is the same. Find the maximum number of fruits per basket.
   (Hint: Dividing two totals into equal groups. Calculate the HCF of 990 and 945.)
7. A gardener has 44 apple trees, 66 banana trees, and 110 mango trees. He wants to plant them in rows such that each row contains only one type of tree and all rows have an equal number of trees. Find the maximum number of trees per row.
   (Hint: Common equal rows for three different totals. Calculate the HCF of 44, 66, and 110.)
8. Find the longest tape which can be used to measure exactly the lengths 336 cm, 240 cm, and 96 cm.
   (Hint: Finding the greatest common divisor of lengths. Calculate the HCF.)
9. A stack of 144 cartons of Coke and 90 cartons of Pepsi are to be arranged in a warehouse. Each stack must have the same number of cartons and be of the same brand. What is the maximum number of cartons possible in each stack?
   (Hint: Equal common stacking. Calculate the HCF of 144 and 90.)
10. A rectangular courtyard is 18m 72cm long and 13m 20cm broad. It is to be paved with square tiles of the same size. Find the side of the largest possible tile.
    (Hint: Converting dimensions to cm and finding the greatest common square side. Calculate the HCF of 1872 and 1320.)
11. 105 goats, 140 donkeys, and 175 cows have to be taken across a river. There is only one boat which can carry the same number of animals of the same kind in each trip. Find the maximum number of animals the boat can carry in one trip.
    (Hint: Finding a common equal group size. Calculate the HCF of 105, 140, and 175.)
12. A shopkeeper has 120 kg of wheat and 150 kg of rice. He wants to pack them into bags of equal weight such that no grain is wasted. What is the maximum weight of each bag?
    (Hint: Dividing totals into equal weights. Calculate the HCF of 120 and 150.)
13. Two rolls of electric wire are 140m and 160m long. They are to be cut into pieces of equal length for a project. What is the greatest possible length of each piece?
    (Hint: Cutting lengths into equal common parts. Calculate the HCF of 140 and 160.)
14. A teacher wants to distribute 100 pens and 125 pencils equally among the maximum number of students possible such that each student gets the same number of pens and the same number of pencils. How many students are there?
    (Hint: Finding the greatest common divisor of the items. Calculate the HCF of 100 and 125.)
15. In a Science Fair, 336 projects from English schools and 240 projects from Science schools are to be arranged in stacks of equal height. Find the maximum projects per stack.
    (Hint: Equal common grouping. Calculate the HCF of 336 and 240.)
16. Determine the maximum length of a scale that can be used to measure exactly three pieces of cloth of lengths 24m, 32m, and 44m.
    (Hint: Finding the largest common factor of three lengths. Calculate the HCF.)
17. 60 students of Class X and 84 students of Class XI are to be seated in a hall in such a way that each row contains the same number of students and students of only the same class. Find the maximum students per row.
    (Hint: Splitting groups into equal rows. Calculate the HCF of 60 and 84.)
18. A milkman has 45 litres and 75 litres of milk in two different cans. Find the maximum capacity of a measuring jug that can empty both cans in an exact number of times.
    (Hint: Common divisor for two volumes. Calculate the HCF of 45 and 75.)
19. Three logs of wood are 12m, 15m, and 21m long respectively. They are to be cut into planks of the same length. What is the greatest possible length of each plank?
    (Hint: Common equal division. Calculate the HCF of 12, 15, and 21.)
20. A tailor has two pieces of cloth of length 90cm and 120cm. He wants to cut them into strips of equal length. What is the maximum length of each strip?
    (Hint: Greatest common length for cutting. Calculate the HCF of 90 and 120.)
21. A jeweler has two gold wires of length 18cm and 24cm. He needs to cut them into equal pieces without wasting any wire. What is the longest possible piece he can cut?
    (Hint: HCF of 18 and 24.)
22. 48 students of Red House and 60 students of Blue House are to be arranged in rows for a drill. Each row must have an equal number of students from the same house. Find the maximum students per row.
    (Hint: Common row size. Calculate the HCF of 48 and 60.)
23. A beverage company wants to fill 120 litres of Orange juice and 180 litres of Apple juice into bottles of equal size. What is the maximum possible size of each bottle?
    (Hint: Common volume divisor. Calculate the HCF of 120 and 180.)
24. 72 boys and 90 girls are to be divided into teams for a quiz. Each team must have the same number of members and consist of either only boys or only girls. Find the maximum team size.
    (Hint: Equal common grouping. Calculate the HCF of 72 and 90.)
25. An event planner has 120 Indian delegates and 180 Foreign delegates. They must be seated in rooms such that every room has the same number of delegates and all delegates in a room are of the same nationality. Find the maximum delegates per room.
    (Hint: Equal grouping logic. Calculate the HCF of 120 and 180.)

26. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they change simultaneously at 7:00 AM, at what time will they change simultaneously again?
    (Hint: Finding the next common multiple of intervals. Calculate the LCM of 48, 72, and 108.)
27. Sonia takes 18 minutes to drive one round of a circular sports field, while Ravi takes 12 minutes for the same. If 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?
    (Hint: Finding the first common time their cycles align. Calculate the LCM of 18 and 12.)
28. Three bells in a temple toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what minimum time will they next toll together?
    (Hint: Finding the next common time. Calculate the LCM of 9, 12, and 15.)
29. Two electronic devices beep after every 60 seconds and 62 seconds respectively. They beeped together at 10:00 AM. At what time will they next beep together?
    (Hint: Common future event time. Calculate the LCM of 60 and 62.)
30. Find the least number of plants a gardener must have so that they can be arranged in rows of 12, 15, or 18 plants each without any plants being left over.
    (Hint: Smallest common total for different row sizes. Calculate the LCM of 12, 15, and 18.)
31. Three people step off together for a morning walk. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance each should walk so that all cover the same distance in complete steps?
    (Hint: Finding the first common distance they meet. Calculate the LCM of 80, 85, and 90.)
32. Two buses leave a depot every 20 minutes and 30 minutes respectively. If they both leave together at 8:00 AM, when is the next time they will leave the depot together?
    (Hint: Finding the next synchronization point. Calculate the LCM of 20 and 30.)
33. Hotdog buns come in packs of 8, and sausages come in packs of 12. What is the least number of packs of each you must buy to have an equal number of buns and sausages?
    (Hint: Finding the smallest common total. Calculate the LCM of 8 and 12, then divide by pack sizes.)
34. A neon sign flashes every 4 seconds, and another sign flashes every 6 seconds. If they flash together now, after how many seconds will they flash simultaneously again?
    (Hint: Future common event. Calculate the LCM of 4 and 6.)
35. Find the smallest number which is exactly divisible by both 28 and 32.
    (Hint: Calculating the least common multiple. Find the LCM.)
36. Three alarm clocks are set to ring every 15, 30, and 45 minutes respectively. If they ring together now, how long will it take for them to ring together again?
    (Hint: Common future interval. Calculate the LCM of 15, 30, and 45.)
37. Tiles of size 20cm by 30cm are used to pave a floor such that they form a perfect square area. What is the smallest possible area of such a square?
    (Hint: The side of the square must be a multiple of both 20 and 30. Calculate the LCM.)
38. A radio station plays a jingle every 15 minutes and a local advertisement every 25 minutes. If they both play at 12:00 PM, at what time will they next play at the same time?
    (Hint: Repeating intervals. Calculate the LCM of 15 and 25.)
39. Two athletes run around a track in 40 seconds and 50 seconds respectively. After how many seconds will they next meet at the starting point?
    (Hint: Synchronizing lap times. Calculate the LCM of 40 and 50.)
40. Find the minimum number of books required for a library so that they can be distributed equally among students of Section A (32 students) or Section B (36 students).
    (Hint: Smallest common total. Calculate the LCM of 32 and 36.)
41. Two drip irrigation emitters drip water every 4 seconds and 10 seconds. After how many seconds will they drip simultaneously?
    (Hint: Time synchronization. Calculate the LCM of 4 and 10.)
42. Ship A sails every 10 days, and Ship B sails every 14 days. If they both sail today, after how many days will they sail together next?
    (Hint: Common future day. Calculate the LCM of 10 and 14.)
43. Two cycle wheels have circumferences of 40cm and 50cm. Find the least distance they must cover to complete a whole number of rotations together.
    (Hint: Finding the common distance multiple. Calculate the LCM of 40 and 50.)
44. Two bells ring at 12:00 PM. If one rings every 18 minutes and the other rings every 24 minutes, what is the next time they will ring together?
    (Hint: Finding the LCM and adding it to 12:00.)
45. What is the least number of marbles that can be grouped into bags of 10, 15, or 25 marbles without any marbles left over?
    (Hint: Smallest common total. Calculate the LCM of 10, 15, and 25.)
46. Ferry A arrives every 45 minutes, and Ferry B arrives every 60 minutes. If they arrive together now, after how many minutes will they arrive together again?
    (Hint: Common interval. Calculate the LCM of 45 and 60.)
47. You go to the gym every 4 days, and your friend goes every 6 days. If you both meet today, after how many days will you both be at the gym together again?
    (Hint: Finding common days. Calculate the LCM of 4 and 6.)
48. A washing machine has a quick cycle of 40 minutes and a heavy cycle of 50 minutes. If both start together, after how many minutes will they finish at the same time?
    (Hint: Synchronization. Calculate the LCM of 40 and 50.)
49. Event A occurs every 4 years and Event B occurs every 6 years. If they coincide in 2024, in what earliest year will they coincide again?
    (Hint: Common multiple of years. Calculate the LCM of 4 and 6.)
50. In a dance routine, Step A occurs every 4 beats and Step B occurs every 6 beats. After how many beats will both steps occur together?
    (Hint: Finding common beats. Calculate the LCM of 4 and 6.)

51. Find the largest number that divides 2053 and 967, leaving remainders 5 and 7 respectively.
    (Hint: Subtract remainders first (2053-5 and 967-7), then calculate the HCF of the results.)
52. Find the least number which when divided by 35, 56, and 91 leaves a remainder of 7 in each case.
    (Hint: Find the smallest common multiple (LCM) and add the “extra” remainder 7.)
53. Determine the greatest number that will divide 445, 572, and 699 leaving remainders 4, 5, and 6 respectively.
    (Hint: Subtract remainders from each number, then find the HCF of the results.)
54. Find the least number which when divided by 20, 25, 35, and 40 leaves remainders 14, 19, 29, and 34 respectively.
    (Hint: Notice that the difference between the divisor and remainder is constant (6). Calculate the LCM and subtract 6.)
55. Find the greatest 4-digit number which is exactly divisible by 15, 24, and 36.
    (Hint: Find the LCM of 15, 24, and 36. Divide 9999 by this LCM and subtract the remainder to find the greatest multiple.)
56. Find the smallest 5-digit number which is exactly divisible by 18, 24, and 30.
    (Hint: Find the LCM. Calculate the first multiple of this LCM that exceeds 9999.)
57. Find the least number which when divided by 6, 7, 8, 9, and 12 leaves the same remainder 1 in each case.
    (Hint: Find the LCM and add 1.)
58. Find the largest number that divides 1251, 9377, and 15628 leaving remainders 1, 2, and 3 respectively.
    (Hint: Subtract remainders and find the HCF.)
59. Find the least number which when divided by 16, 18, and 20 leaves a remainder of 4 in each case.
    (Hint: Find the LCM and add 4.)
60. Find the greatest number which divides 615 and 963 leaving a remainder of 6 in each case.
    (Hint: Find HCF of 609 and 957.)
61. What is the smallest number that leaves a remainder of 3 when divided by 12, 15, and 18?
    (Hint: Find the LCM and add 3.)
62. Find the largest number that divides 546 and 764 leaving remainders 6 and 8 respectively.
    (Hint: Find HCF of 540 and 756.)
63. Find the smallest number of 4 digits divisible by 12, 15, 18, and 27.
    (Hint: Find the LCM and find its smallest 4-digit multiple.)
64. Find the least number which when divided by 12, 16, 24, and 36 leaves a remainder of 7 in each case.
    (Hint: Find the LCM and add 7.)
65. The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other.
    (Hint: Use the formula \(HCF \times LCM = Product\ of\ two\ numbers\).)
66. Find the least number which when increased by 15 is exactly divisible by 15, 35, and 48.
    (Hint: Find the LCM of 15, 35, 48 and then subtract 15.)
67. Find the least number which when decreased by 7 is divisible by 12, 16, 18, 21, and 28.
    (Hint: Find the LCM and add 7.)
68. Find the largest number dividing 398, 436, and 542 leaving remainders 7, 11, and 15 respectively.
    (Hint: Subtract the remainders from each number and find the HCF.)
69. Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
    (Hint: Difference is constant (20). Find the LCM and subtract 20.)
70. Find the largest number that divides 245 and 1029 leaving a remainder of 5 in each case.
    (Hint: Subtract 5 from both and find the HCF.)
71. Find the smallest number divisible by each of the first 10 natural numbers (1 to 10).
    (Hint: Calculate the LCM of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.)
72. Two numbers are in the ratio 15:11. If their HCF is 13, find the numbers.
    (Hint: Let numbers be \(15x\) and \(11x\). Since they are in simplest ratio, \(x\) must be the HCF. Multiply 13 by 15 and 11.)
73. Can two numbers have 16 as their HCF and 380 as their LCM? Give reasons.
    (Hint: For any two numbers, the LCM must be exactly divisible by the HCF. Check if 380/16 is an integer.)
74. The sum of two numbers is 528 and their HCF is 33. How many pairs of such numbers exist?
    (Hint: Numbers are \(33a\) and \(33b\). \(33(a+b) = 528 \implies a+b = 16\). Find co-prime pairs.)
75. The product of two numbers is 12960 and their HCF is 18. Find their LCM.
    (Hint: Use \(Product / HCF = LCM\).)

76. A hall is 25m 11cm long and 13m 69cm broad. (a) Find the size of the largest square tile to pave it. (b) Find the least number of square tiles required to pave the floor.
    (Hint: Find the HCF for the tile side. Then, Number of Tiles = Total Area / Area of 1 Tile.)
77. Three bells toll at intervals of 12, 15, and 18 seconds respectively. If they toll together for the first time, how many times will they toll together in 20 minutes?
    (Hint: Find the LCM. Divide 1200 seconds by the LCM.)
78. A hotel has 120 Indian and 180 Foreign delegates. They are to be seated in rooms such that each room has the same number of delegates and same nationality. (a) Find the maximum per room. (b) Find the total number of rooms required.
    (Hint: (a) Find HCF. (b) Total Delegates / HCF.)
79. Runners A, B, and C take 6, 8, and 10 minutes respectively to complete one lap of a track. (a) After how much time will they meet at the start? (b) How many laps will Runner A have completed by then?
    (Hint: (a) Find the LCM. (b) Divide the LCM by 6.)
80. A farmer has 945 cows and 2475 sheep. He forms them into flocks of equal size such that only one kind of animal is in a flock. (a) Max size per flock? (b) Find the total number of flocks.
    (Hint: (a) Find HCF. (b) (945+2475) / HCF.)
81. Lamps A, B, and C flash every 6s, 9s, and 12s. (a) When do they sync? (b) How many times do they flash together in exactly one hour?
    (Hint: Sync = LCM. 3600 / LCM.)
82. Books of English (336), Science (240), and Maths (96) are stacked equally. (a) Find the max books per stack. (b) Find the total number of stacks.
    (Hint: (a) Find HCF. (b) Total books / HCF.)
83. Three measuring rods are 64cm, 80cm, and 96cm. (a) Find the minimum length of cloth measured exactly by all. (b) How many times is each rod used for this length?
    (Hint: (a) Find LCM. (b) Divide LCM by each rod length.)
84. A gift hamper contains 156 chocolates, 208 candies, and 260 biscuits. (a) Max identical hampers? (b) How many items of each kind are in one hamper?
    (Hint: (a) Find HCF. (b) Divide each quantity by the HCF.)
85. Two towers are built of 15cm and 25cm blocks respectively. (a) What is the minimum equal height both towers can reach? (b) How many blocks of each kind are used?
    (Hint: (a) Find LCM. (b) Height / block size.)
86. Find the HCF of 210 and 55 and express it in the form of \(210x + 55y\).
    (Hint: Find HCF using Euclid’s Algorithm, then work backwards.)
87. Find the greatest 3-digit number which is exactly divisible by 8, 10, and 12.
    (Hint: Find the LCM and find its largest 3-digit multiple.)
88. What is the least number of soldiers in a parade that can be arranged in rows of 15, 20, and 25?
    (Hint: Find LCM.)
89. Determine the smallest number which when divided by 12, 15, 20, and 54 leaves a remainder of 8 in each case.
    (Hint: Find the LCM and add 8.)
90. The LCM of two numbers is 14 times their HCF. The sum of the LCM and HCF is 600. If one number is 80, find the other.
    (Hint: Let HCF be \(x\). \(x + 14x = 600\). Find \(x\), then use the product rule.)
91. Find the least square number which is exactly divisible by each of the numbers 6, 9, 15, and 20.
    (Hint: Find the LCM. Prime factorize it and multiply by missing factors to make all exponents even.)
92. Runners A, B, and C run on a circular track of 12km at speeds of 3, 7, and 13 km/hr. When will they meet at the start?
    (Hint: Time = Distance/Speed. Find LCM of fractions \(12/3, 12/7, 12/13\). LCM of fractions = LCM of Numerators / HCF of Denominators.)
93. Find all pairs of positive integers whose sum is 91 and whose HCF is 7.
    (Hint: \(7a + 7b = 91 \implies a + b = 13\). List co-prime pairs \((a,b)\).)
94. The HCF of two numbers is 18 and their product is 12960. Find the numbers if they are in the ratio 5:8.
    (Hint: Numbers are \(5x\) and \(8x\). \(5x \times 8x = 12960 \implies 40x^2 = 12960\). Solve for \(x\).)
95. A number when divided by 10, 9, and 8 leaves remainders 9, 8, and 7 respectively. Find the smallest such number.
    (Hint: Common difference (Divisor – Remainder) is 1. Find LCM and subtract 1.)
96. Determine the smallest number which when divided by 24, 36, and 48 leaves remainders 21, 33, and 45 respectively.
    (Hint: Common difference is 3. Calculate LCM and subtract 3.)
97. The HCF of 408 and 1032 is expressible in the form \(1032m – 408 \times 5\). Find the value of \(m\).
    (Hint: Find the HCF using Euclid’s Algorithm, equate it to the expression, and solve for \(m\).)
98. Find the least number of square tiles required to pave a room that is 15m 17cm long and 9m 2cm broad.
    (Hint: HCF gives the tile side. Then, Total Area / Tile Area.)
99. The sum of two numbers is 105 and their LCM is 180. Find the numbers.
    (Hint: HCF divides both Sum and LCM. Use \(H=HCF(105, 180)\).)
100. Find HCF and LCM of 12, 15, and 21. Verify whether the product of the three numbers is equal to the product of their HCF and LCM.
     (Hint: Calculate and check. For 3 numbers, this is false.)