Maharashtra Board Class 6 Maths part 1 Chapter 9 HCF LCM PDF Download

HCF-LCM

Divisor, Dividend

In the empty boxes, write the proper words: dividend, divisor, quotient, remainder

When we divide 36 by 4, the remainder is zero. Therefore, 4 is a factor of 36 and 36 is a multiple of 4 (36 is divisible by 4).

When we divide 65 by 9, the remainder is not zero. Therefore, 9 is not a factor of 65. Also, 65 is not a multiple of 9 (65 is not divisible by 9).

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Write the common factors: 1, 2, 3, 4, 6, 12

Practice Set 23

Write all the factors of the given numbers and list their common factors:

(1) 12, 16

(2) 21, 24

(3) 25, 30

(4) 24, 25

(5) 56, 72

Highest Common Factor: HCF

Example: My aunt has brought two ribbons of different colours, one 12 metres long and the other 18 metres. Both the ribbons have to be cut into pieces of the same length. What should be the maximum length of each piece?

The number which gives the length of each piece must be a factor of 12 and of 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Of the common factors of 12 and 18, 6 is the greatest. Therefore, the maximum length of each piece should be 6 metres.

Teacher's Note

HCF helps us divide things equally. For example, if you have 12 apples and 18 oranges, you can give 6 apples and 6 oranges to each friend.

Exam Trick

Remember: HCF means the biggest number that divides all numbers. Write all factors and circle the biggest common one.

Points to Remember

HCF is the Highest Common Factor.
HCF divides all the given numbers.
The HCF is always the biggest common factor.
To find HCF, list all factors and pick the biggest one.
HCF helps divide things into equal parts.

Page 2

Example: There are 20 kg of jowar and 50 kg of wheat in a shop. All the grain is to be packed in bags. If all the bags are to have equal weights of grain, what is the maximum weight of grain that can be filled in each bag?

The weight of the grain in each bag must be a factor of 20 and 50. Besides, the maximum possible weight must be filled in each bag. Hence, let us find the HCF of 20 and 50.

Factors of 20: 1, 2, 4, 5, 10, 20

Factors of 50: 1, 2, 5, 10, 25, 50

Common factors: 1, 2, 5, 10

Of the common factors of 20 and 50, 10 is the greatest. That is, 10 is the HCF of the numbers 20 and 50.

Therefore, a maximum of 10 kg of grain can be filled in each bag.

To find the HCF of given numbers, we make a list of the factors of the numbers and find the greatest of the common factors.

Practice Set 24

1. Find the HCF of the following numbers:

(1) 45, 30

(2) 16, 48

(3) 39, 25

(4) 49, 56

(5) 120, 144

(6) 81, 99

(7) 24, 36

(8) 25, 75

(9) 48, 54

(10) 150, 225

2. If large square beds of equal size are to be made for planting vegetables on a plot of land 18 metres long and 15 metres wide, what is the maximum possible length of each bed?

3. Two ropes, one 8 metres long and the other 12 metres long are to be cut into pieces of the same length. What will the maximum possible length of each piece be?

4. The number of students of Std 6th and Std 7th who went to visit the Tadoba Tiger Project at Chandrapur was 140 and 196 respectively. The students of each class are to be divided into groups of the same number of students. Each group can have a paid guide. What is the maximum number of students there can be in each group? Why do you think each group should have the maximum possible number of students?

5. At the Rice Research Centre at Tumsar, there are 2610 kg of seeds of the basmati variety and 1980 kg of the Indrayani variety. If the maximum possible weight of seeds has to be filled to make bags of equal weight, what should be the weight of each bag? How many bags of each variety will there be?

Teacher's Note

HCF is useful in real life. In your house, if you want to pack 20 mangoes and 30 guavas into boxes equally, HCF helps you find the right box size.

Exam Trick

For HCF word problems, always ask: "What divides both numbers equally?" Write factors and pick the biggest common one.

Points to Remember

HCF solves sharing and packing problems.
List all factors of each number carefully.
Circle only the common factors.
The biggest common factor is the HCF.
Use HCF when you need equal groups or equal sizes.

Lowest Common Multiple: LCM

Write down the 3 times and 4 times tables. Note that a table has the multiples of a number arranged in serial order. Which is the smallest number divisible by both 3 and 4?

We find the lowest common multiple or the LCM useful for certain purposes in daily life. Can you find the greatest common multiple of some given numbers? Think about it.

Rehana and Anne are making garlands of flowers. Both have to be given an equal number of flowers in their baskets.

Tai: Rehana, you make garlands of 6 flowers each. Anne, you make garlands of 8 flowers each. What is the minimum number of flowers I should put in your baskets?

Rehana: I want flowers in multiples of 6.

Anne: I want them in multiples of 8.

Multiples of 6 are numbers that are divisible by 6 and they are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78,

Multiples of 8 are numbers that are divisible by 8 and they are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104,

The common multiples are 24, 48, 72, 96,

Rehana: Tai, if you give each of us 24, 48, 72 or 96 flowers, we will both be able to make the garlands as you want.

Anne: You will have to give us at least 24 flowers.

24 is the lowest common multiple or LCM of 6 and 8.

Example: Find the LCM of 13 and 6.

13 times table: 13, 26, 39, 52, 65, 78, 91, 104, 117, 130

6 times table: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Here, we see no common multiples. So, let us extend the tables.

Further numbers divisible by 13: 130, 143, 156,

Further numbers divisible by 6: 60, 66, 72, 78, 84,

Looking at the lists of numbers divisible by 13 and 6, we see that 78 is the lowest common multiple. Therefore, the LCM of 13 and 6 is 78.

The LCM of two numbers cannot be bigger than their product.

Teacher's Note

LCM is useful when things happen again at the same time. For example, if a bus comes every 6 minutes and a train every 8 minutes, LCM tells you when both come together.

Exam Trick

Remember: LCM = smallest number in both times tables. Write multiples of each number and find the smallest one that appears in both lists.

Points to Remember

LCM is the Lowest Common Multiple.
LCM is divisible by all the given numbers.
The LCM is the smallest common multiple.
To find LCM, write multiples and pick the smallest common one.
LCM tells us when things happen together again.

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