Maharashtra Board Class 6 Maths Chapter 9 HCF LCM Set 25 Solutions

Question 1. Find out the LCM of the following numbers.
(i) 9,15
(ii) 2,3,5
(iii) 12,28
(iv) 15,20
(v) 8,11
Answer:
(i) Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
Multiples of 15 = 15, 30, 45
∴ LCM of 9 and 15 = 45
(ii) Multiples of 2 = 2, 4,6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 5 = 5, 10, 15, 20, 25, 30
∴ LCM of 2,3 and 5 = 30
(iii) Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Multiples of 28 = 28, 56, 84
∴ LCM of 12 and 28 = 84
(iv) Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120
Multiples of 20 = 20, 40, 60
∴ LCM of 15 and 20 = 60
(v) Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96
Multiples of 11 = 11, 22, 33, 44, 55, 66, 77, 88
∴ LCM of 8 and 11 = 88
In simple words: The Least Common Multiple (LCM) is the smallest positive integer that is divisible by each of the given integers. To find the LCM, list multiples of each number until you find the first common multiple.

🎯 Exam Tip: Always list out enough multiples to clearly identify the smallest common one. For prime numbers or numbers with no common factors, their LCM is simply their product.

Question 2. Solve the following problems:
(i) On the playground, if the children are made to stand for drill either 20 to a row or 25 to a row, all rows are complete and no child is left out. What is the lowest possible number of children in that school?
(ii) Veena has some beads. She wants to make necklaces with an equal number of beads in each. If She makes necklaces of 16 or 24 or 40 beads, there is no bead left over. What is the least number of beads with her?
(iii) An equal number of laddoos have been placed in 3 different boxes. The laddoos in the first box were distributed among 20 children equally, the laddoos in the second box among 24 children and those in the third box among 12 children. Not a single laddoo was left over. What was the minimum number of laddoos in the three boxes altogether?
(iv) We observed the traffic lights at three different squares on the same big road. They turn green every 60 seconds, 120 seconds and 24 seconds. When the signals are switched on at 8 o'clock in the morning, all the lights were green. How long after that will all three signals turn green simultaneously again?
(v) Given the fractions \( \frac{13}{45} \) and \( \frac{22}{75} \). Write their equivalent fractions with same denominators and add the fractions.
Answer:
(i) The lowest possible number of children is equal to the lowest common multiple of 20 and 25.
Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200
Multiples of 25 = 25, 50, 75, 100
∴ LCM of 20 and 25 = 100
∴ The least number of students in the school is 100.

(ii) The least number of beads with Veena is equal to the lowest common multiple of 16, 24 and 40.
Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288
Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, 240
Multiples of 40 = 40, 80, 120, 160, 200, 240
∴ LCM of 16, 24 and 40 = 240
∴ The least number of beads with Veena are 240.

(iii) The lowest common multiple of 20, 24 and 12 gives the minimum number of laddoos in one box.
Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200
Multiples of 24 = 24, 48, 72, 96, 120
Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
∴ LCM of 20, 24 and 12 = 120
∴ Minimum number of ladoos in 1 boxes = 120
∴ Minimum number of ladoos in 3 boxes = 3 x 120 = 360
∴ The minimum number of ladoos in 3 boxes are 360.

(iv) All three signals will turn green for lowest common multiple of 60 seconds, 120 seconds and 24 seconds.
Multiples of 60 = 60, 120, 180, 240, 300, 360, 420, 480
Multiples of 120 = 120, 240, 360
Multiples of 24 = 24, 48, 72, 96, 120
LCM of 60, 120 and 24 = 120
Since, 60 seconds = 1 minute
∴ 120 seconds = 2 minutes
∴ The signals will turn green simultaneously again after 120 seconds i.e. 2 minutes.

(v) The lowest common multiple of 45 and 75 gives the same denominator.
Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450
Multiples of 75 = 75, 150, 336
∴ LCM of 45 and 75 = 225
\( \frac{13}{45} = \frac{13 \times 5}{45 \times 5} = \frac{65}{225} \)
\( \frac{22}{75} = \frac{22 \times 3}{75 \times 3} = \frac{66}{225} \)
\( \therefore \frac{13}{45} + \frac{22}{75} = \frac{65}{225} + \frac{66}{225} = \frac{65+66}{225} = \frac{131}{225} \)
\( \therefore \) Equivalent fractions of \( \frac{13}{45} \) and \( \frac{22}{75} \) are \( \frac{65}{225} \) and \( \frac{66}{225} \) respectively, and the addition of \( \frac{13}{45} \) and \( \frac{22}{75} \) is \( \frac{131}{225} \)
In simple words: These problems all require finding the Least Common Multiple (LCM) to determine the earliest common point in time or the smallest common quantity. By listing multiples or using prime factorization, the LCM helps identify when events will coincide or how many items are needed to satisfy multiple conditions.

🎯 Exam Tip: Word problems involving "least number," "minimum," "smallest," or "when will they next happen together" usually indicate that you need to find the LCM. Clearly show your steps for finding the LCM for full marks.

Maharashtra Board Class 6 Maths Chapter 9 HCF-LCM Practice Set 25 Intext Questions and Activities

Question 1. Pravin, Bageshri and Yash are cousins who live in the same house. Pravin is an Army Officer. Bageshri is studying in a Medical College in another city. Yash lives in a nearby town in a hostel. Pravin can come home every 120 days. Bageshri comes home every 45 days and Yash, every 30 days. All three of them left home at the same time on the 15th of June 2016. Their parents said, "We shall celebrate like a festival the day you all come home together.” Mother asked Yash, "What day will that be?" Yash said, "The number of days after which we come back together must be divisible by 30, 120. That means we shall be back together on the 10th of June next year. That will certainly be a for us!" How did Yash find the answer? (Textbook pg. no. 49)
ℹ️ चित्र व्याख्या (Diagram Explanation): एक चित्र जिसमें एक सेना अधिकारी वर्दी में अपने परिवार के सदस्यों से बात कर रहा है, जिसमें एक लड़का स्कूल बैग लिए हुए है और दो महिलाएं हैं। यह परिवार के विभिन्न सदस्यों के साथ रहने की कहानी को दर्शाता है।
Answer:
The day when Pravin, Bageshri and Yash come back together is lowest common multiple of 30, 45 and 120.
Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360
Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360
Multiples of 120: 120, 240, 360
∴ They will come together after 360 days
Day when they left home = 15th June
∴ Day when they come back together = 15th June + 360 days
= 10th June next year
∴ Pravin, Bageshri and Yash will come back together on 10th June next year.
In simple words: Yash found the answer by calculating the Least Common Multiple (LCM) of the days each person returns home (30, 45, and 120 days). The LCM represents the minimum number of days after which all three will return simultaneously. Adding this number of days to their departure date reveals the exact reunion date.

🎯 Exam Tip: This problem is a classic application of LCM. Ensure you accurately list multiples or use prime factorization for all numbers involved to find the correct LCM. Don't forget to convert the total days back into a calendar date.

Question 2. A Maths Riddle! (Textbook pg. no. 50)
We have four papers. On each of them there is a number on one side and some information on the other. The numbers on the papers are 7, 2, 15, 5. The information on the papers is given below in random order.
(i) A number divisible by 7
(ii) A prime number
(iii) An odd number
(iv) A number greater than 100
If the number on every paper is mismatched with the information on its other side, what is the number on the paper which says 'A number greater than 100?
Answer:

Based on the mismatches above:
- Paper '2' has "An odd number" (iii).
- Paper '5' has "A number divisible by 7" (i).
- Paper '15' has "A prime number" (ii).
The remaining number is '7' and the remaining information is "A number greater than 100" (iv). Since every paper is mismatched, '7' must be mismatched with "A number greater than 100".
Therefore, the number on the paper which says 'A number greater than 100' is 7.
In simple words: This riddle requires a process of elimination based on the rule that every number is mismatched with its description. By systematically pairing numbers with the descriptions they *cannot* fit, we can deduce the correct mismatched pairing for the final number and description.

🎯 Exam Tip: For logic-based problems like this, create a clear mapping or table to track possibilities and eliminations. This helps visualize the mismatches and arrive at the correct conclusion without errors.