__L.C.M & H.C.F__**Factors and Multiples: If a number' divides another number `n' exactly, then we say that',**

m' is a factor of `n' and that `n' is a multiple of `m'.neg. 3 is a factor of 12 and therefore 12 is a multiple of 3.

__Least Common Multiple (L.C.M.)__
L.C.M. is the least non-zero number in common multiples of two or more numbers.

Multiple of 6 = 6, 12, 18, 24, 30, ........

Multiple of 8 = 8, 16, 24, 32, 40, ........

Common Multiple of 6 and 8 = 24, 48

Least Common Multiple = 24

__Factorisation Method__:
Find the L.C.M. of 12, 27 and 40

Factors of 12

= 2x2x3 = 2

^{2}x3
Factors of 27

= 3x3x3 = 3

^{3}
Factors of 40

= 2x2x2x5 = 2

^{3}x5
L.C.M. = 2

^{3}x 3^{3}x5 = 10 8 0**SHORTCUT METHOD**

(Division Method)

**Find the L.C.M. of 12, 27, 40**

L.C.M. =2x2x3x9x10 = 1080

__HIGHEST COMMON FACTOR (H.C.F)__
The highest common factor of two or more numbers is the greatest number which divides each of them exactly. eg. Find the H.C.F. of 24 and 56

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56

Common factors of 24 and 56 are 1, 2, 4, 8

H.C.F. of 24 and 56 = 8

__Factorisation Method:__
H.C.F. can be found by resolving the given numbers into prime factors and then taking the product of least powers of all common factors, that occur in these numbers.

Eg. Find H.C.F. of 48, 108, 140

Factors of 48

= 2x2x2x2x3

= 24x3

Factors of 108

= 2x2x3x3x3 = 2

^{2}x3^{3}
Factors of 140

= 2x2x5x7 = 2

^{2}x 5 x 7
H.C.F. = 2

^{2}= 4

**Division Method**

Find the H.C.F. of 48, 108, 140

H.C.F.=2x2= 4

__QUICKER & SHORTCUT METHOD__

Find the H.C.F. of 777 and 1147

H.C.F. of 777 and 1147 is 37

* The product of two given numbers is equal to the product of their H.C.F. and L.C.M.

The L.C.M of a given set of numbers would be either the highest or higher than the highest of the given numbers.

The H.C.F. of a given set of numbers would be either the lowest or lower than the lowest

**Solved Examples**

1. Find the L.C.M. of 125,64,8 and 3.

Ans: Given numbers are 53, 26, 23 and 3

L.C.M. 53x26x3 = 24,000

2. Find the L.C.M. of

1/3, 5/6,5/9,10/27?

Ans: L.C.M. of fractions =

__L CM of numerators__
H C F of denominators

L.C.M. of 1, 5 and 10 is 10

H.C.F of 3, 6, 9 and 27 is 3

L.C.M. of given fractions =10/3

3. Find the H.C.F. of 1/2,3/4,5/6,7/8,9/10

Ans: H.C.F. of fractions =

__H C F of numerators__
LCM of denominators

H.C.F. of 1, 3, 5, 7 and 9 is 1

L.C.M of 2, 4, 6, 8 and 10 is 120

H.C.F. of given fractions =1/120

4 . The L.C.M. of two number is 2310. Their H.C.F. is 30. If one number is 210, the other is:

Ans: The other number =

given number

=

210

= 330

5. The H.C.F. and L.C.M. of two numbers are 44 and 264 respectively. If the first number is divide 2, the quotient is 44, The other number is:

__LCM x H C F__given number

=

__2310 * 30__210

= 330

5. The H.C.F. and L.C.M. of two numbers are 44 and 264 respectively. If the first number is divide 2, the quotient is 44, The other number is:

Ans: First number = 2x44 = 88

Second number =

Second number =

__44 x 264__
88

=132

6. The least square number which is divisible by 6, 8 and 15 is:

Ans: The least number divisible by 6, 8 and 15 is their L.C.M. which is 120

Now 120 = 2x2x2x3x5

To make it a perfect square, it must be multiplied by 2x3x5 Required Number=120x2x3x5=3600

=132

6. The least square number which is divisible by 6, 8 and 15 is:

Ans: The least number divisible by 6, 8 and 15 is their L.C.M. which is 120

Now 120 = 2x2x2x3x5

To make it a perfect square, it must be multiplied by 2x3x5 Required Number=120x2x3x5=3600

7. The least number of square tiles required to pave the ceiling of a room 15m 17cm long and 9m. 2cm broad is:

Ans: Size of largest square tile

= H.C.F. of 1517 cm and 902 cm = 41 cm.

Least number of tiles required = Area of the room / Area of one tile

Ans: Size of largest square tile

= H.C.F. of 1517 cm and 902 cm = 41 cm.

Least number of tiles required = Area of the room / Area of one tile

=

41 x 41

= 814

__1517 x 902__41 x 41

= 814

**PRACTICE TEST**
1 . Find the L.C.M of 12, 15, 18 and 27.

a) 1, 080 b) 5 4 0 c) 2 7 0 d) 7 6 0

a) 1, 080 b) 5 4 0 c) 2 7 0 d) 7 6 0

2 . Find the H.C.F. of 72, 48 and 30.

a) 3 0 b) 1 2 c) 6 d) 3

3 . Find the L.C.M. of 22x33x53 and 23x32x5.

a) 27,000 b) 1 8 0 c) 3 6 d) 13,500

4. H.C.F of two numbers is 24 and their L.C.M is 1080. If one of the numbers is 120, find the other.

a) 2 1 6 b) 5 3 2 c) 1 0 8 d) 8 2 0

5 . L.C.M. of 2.5, 0.5 and 0.175 = ?

a) 2. 5 b) 0. 5 c) 0. 175 d) 17. 5

6 . H.C.F. of two numbers is 24 and their L.C.M is 1344. If the difference between the numbers is 80, their sum is:

a) 3 6 8 b) 3 5 6 c) 3 3 2 c) 3 0 4

7. Find the greatest number which can divide 1354, 1866 and 2762 leaving the same

remainder 10 in each case. a) 6 4 b) 1 2 4 c) 1 5 6 d) 2 6 0

8. Two containers contain 60 and 165 litres of milk respectively. Find the maximum capacity of a container which can measure the milk in each container an exact number of times

(in litres) a) 1 5 b) 3 c) 5 d) 1 0

9. Two baskets contain 195 and 250 bananas respectively, which are distributed in equal number among children. Find the largest number of bananas that can be given, so that 3 bananas are left over from the first basket and 2 from the second. a) 4 b) 1 8 c) 8 d) 6

10. A heap of stones can be made in groups of 21 but when made up into groups of 16, 20, 25 and 45 there are 3 stones left in each case, The number of stones in the heap is

a) 36 00 b) 36 03 c) 72 00 d) 72 03

a) 3 0 b) 1 2 c) 6 d) 3

3 . Find the L.C.M. of 22x33x53 and 23x32x5.

a) 27,000 b) 1 8 0 c) 3 6 d) 13,500

4. H.C.F of two numbers is 24 and their L.C.M is 1080. If one of the numbers is 120, find the other.

a) 2 1 6 b) 5 3 2 c) 1 0 8 d) 8 2 0

5 . L.C.M. of 2.5, 0.5 and 0.175 = ?

a) 2. 5 b) 0. 5 c) 0. 175 d) 17. 5

6 . H.C.F. of two numbers is 24 and their L.C.M is 1344. If the difference between the numbers is 80, their sum is:

a) 3 6 8 b) 3 5 6 c) 3 3 2 c) 3 0 4

7. Find the greatest number which can divide 1354, 1866 and 2762 leaving the same

remainder 10 in each case. a) 6 4 b) 1 2 4 c) 1 5 6 d) 2 6 0

8. Two containers contain 60 and 165 litres of milk respectively. Find the maximum capacity of a container which can measure the milk in each container an exact number of times

(in litres) a) 1 5 b) 3 c) 5 d) 1 0

9. Two baskets contain 195 and 250 bananas respectively, which are distributed in equal number among children. Find the largest number of bananas that can be given, so that 3 bananas are left over from the first basket and 2 from the second. a) 4 b) 1 8 c) 8 d) 6

10. A heap of stones can be made in groups of 21 but when made up into groups of 16, 20, 25 and 45 there are 3 stones left in each case, The number of stones in the heap is

a) 36 00 b) 36 03 c) 72 00 d) 72 03

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