Story of Prime Factors (Class- 6, Experience- 3)

[Exercise of H.C.F- Page 59]

H.C.F-E.3.1. Through the picture, determine the H.C.F of the following numbers according to the method of division.

(a) 24, 45, 62

(b) 56, 78, 90

(c) 120, 56, 78

(d) 99, 33, 123

(e) 95, 57, 23

Solution:

H.C.F-E.3.2. The H.C.F of 100 and 44 can be determined from the following picture. Can you tell how?

Solution:

H.C.F-E.3.3. There are two ropes of 15 m and 40 m length. Cut these two ropes into small pieces of the same length so that no part of the rope is damaged. What can be the maximum length of the small pieces?

Solution:

H.C.F-E.3.4. A shopkeeper sells candles in both packets of 12 and 8. To have one candle for each candle stand, what is the minimum number of candles and candle stands that Ayesha has to buy?

Solution:

H.C.F-E.3.5. A florist wants to arrange 24 bouquets in different rows. In how many different ways can he arrange them with the same number of bouquets in each row?

Solution:

H.C.F-E.3.6. 210 Oranges, 252 Apples and 294 Pears are evenly packed in cartons so that no fruit is left out. What is the maximum number of cartons that will be needed there?

Solution:

H.C.F-E.3.7. The length, width, and height of a room are 6 m 80 cm, 5 m 10 cm and 3 m 40 cm respectively. You will be given a stick only, not a scale. The length of that stick will be as you want but you can only demand it once. That means you’ll get only one stick. With this stick you have to make sure that the length, width and height of the room are measured accurately. What is the maximum length of stick you can ask for?

Solution:

H.C.F-E.3.8. H.C.F of two numbers is 6, if one number is 42, find the other number?

Solution:

H.C.F-E.3.9. Activity with the help of bucket and water:

a) How to measure 4 liters of water with 3 liter and 5 liter water buckets? In this case, there will be no measuring marks on the bucket. Again, other measuring instrument such as scale or measuring scales etc. cannot be used.

b) Which of the following amount of water can be measured with 4 liter and 6 liter water buckets? (In this case, there will be an opportunity to keep in other containers for 7, 8, 9, 10 liters)

Solution:

a) To measure exactly 4 liters of water using only 3 liter and 5 liter buckets without any measuring marks or additional tools, you can follow these steps:

1. Fill the 3 liter bucket completely.

2. Pour the water from the 3 liter bucket into the 5 liter bucket.

3. Fill the 3 liter bucket again.

4. Carefully pour water from the 3 liter bucket into the 5 liter bucket until it's full. This will leave exactly 1 liter of space remaining in the 5 liter bucket.

5. Now, you have 1 liter of water in the 3 liter bucket.

6. Empty the 5 liter bucket.

7. Pour the remaining 1 liter of water from the 3 liter bucket into the empty 5 liter bucket.

8. Fill the 3 liter bucket again.

9. Pour the water from the 3 liter bucket into the 5 liter bucket until it's full.

At this point, the 5 liter bucket will have 4 liters of water, which is the desired amount.

b) To determine which amounts of water can be measured using only 4 liter and 6 liter buckets, along with additional containers of 7, 8, 9, and 10 liters, we need to consider the combinations of volumes we can create.

First, let's establish what amounts can be directly measured with just the 4 liter and 6 liter buckets:

Since the greatest common divisor of 4 and 6 is 2 liters, we can directly measure any multiple of 2 liters using these buckets. This includes 2 liters, 4 liters, 6 liters, etc.

Now, let's consider combinations with the additional containers:

• If we have a 7-liter container:

We can measure 7 liters by combining 4 liters (from the 4 liter bucket) and 3 liters (from the 6 liter bucket).

• If we have an 8-liter container:

We can measure 8 liters by combining 4 liters (from the 4 liter bucket) and 4 liters (from the 6 liter bucket).

• If we have a 9-liter container:

We can measure 9 liters by combining 3 liters (from the 4 liter bucket) and 6 liters (from the 6 liter bucket).

• If we have a 10-liter container:

We can measure 10 liters by combining 6 liters (from the 4 liter bucket) and 4 liters (from the 6 liter bucket).

Therefore, with the 4 liter and 6 liter buckets, along with additional containers of 7, 8, 9, and 10 liters, we can measure 2, 3, 4, 6, 7, 8, 9, and 10 liters of water.

[Exercise of L.C.M- Page 55]

L.C.M-E.3.1. Determine L.C.M following all the possible ways as discussed in the section ‘Tree of L.C.M’ with the help of tree of prime factors.

(a) 14, 15, 12

(b) 66, 78, 100

(c) 120, 56, 60

(d) 55, 15, 143

(e) 25, 57, 95

Solution:

L.C.M-E.3.2. Prove that ‘the product of two numbers is equal to the product of both the numbers’ H.C.F and L.C.M.’

Solution:

L.C.M-E.3.3. What is the minimum number of students that can be arranged in groups of 3, 4, 6 and 8 so that no one is left out?

Solution:

To find the minimum number of students that can be arranged in groups of 3, 4, 6, and 8 so that no one is left out, we need to find the least common multiple (LCM) of these numbers. The LCM of 3, 4, 6, and 8 can be calculated as follows:

LCM(3, 4, 6, 8)

To calculate this, we find the prime factorization of each number and take the highest power of each prime factor that appears:

• Prime factorization of 3:

3^{1}

• Prime factorization of 4:

2^{2}

• Prime factorization of 6:

2^{1}

3^{1}

• Prime factorization of 8:

2^{3}

The LCM is the product of the highest powers of all prime factors:

LCM(3,4,6,8) =

2^{3}

3^{1}

Therefore, the minimum number of students needed to form groups of 3, 4, 6, and 8 without anyone being left out is 24.

L.C.M-E.3.4. There are 2 types of buses in a local bus service that starts from 8 am. The first type of buses leave after every 15 minutes and the second type of buses leave after every 20 minutes. How many times do the first and second type of buses leave at the same time between 8 am and 11 am on a given day?

Solution:

L.C.M-E.3.5. Three painters, Ron, Habib and Shelley, are designing a hotel room. The hotel has room numbers from 15 to 200. Ron has to work in all the rooms. Habib has to work in the rooms where the room number is a multiple of 3. Shelley has to work in the rooms where the room number is a multiple of 5. In which rooms will they all work together?

Solution:

L.C.M-E.3.6. Rasheda goes to a shopping mall every 6th day in a week. Andy goes to the same shopping mall every 7th day. How many times will they meet each other in the mall in December and January if the counting starts from 1st December?

Solution:

L.C.M-E.3.7. Sami can jump 4 steps at a time and Nina can jump 5 steps at a time. If the two starts jumping together, at what step will they meet?

Solution:

L.C.M-E.3.8. Aumiya has a music class every 2nd day and a painting class every 3rd day. On which day will she have both the classes?

Solution:

L.C.M-E.3.9. Today, both the football team and the basketball teams were playing. The football team plays 3 days in a week and the basketball team plays 5 days in a week. When next the two teams will play on the same day?

Solution:

L.C.M-E.3.10. In the picture two separate piles are being made side by side using two different shaped square boxes. What is the minimum number of orange and blue boxes that will be required to equalize the height of the two piles? What is the minimum height required for the two piles to be equal?

Solution:

L.C.M-E.3.11. In a marathon race, two people start drinking water after starting the race at regular intervals. The first person drinks water in every 9 minutes. 72 minutes after the start of the race, two men drank water at the same time. At what interval does the second person drink water? How many times does the second person drink water in 72 minutes?

Solution:

L.C.M-E.3.12. Bus A and Bus B are two intercity service of Dhaka. Bus A service leaves the bus stand in every 60 minutes and Bus B service leaves the same bus stand in every 80 minutes. Everyday they start their journey at 6 AM. How many times and at what times in a day they leave the bus stand together?

Solution:

Active Math Class

Story of Prime Factors (Class- 6, Experience- 3)

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