Highest Common Factor
The highest common factor of two or more given numbers is
the largest of their common factors. It is known
as GCD also.
eg, Factors of 20 are 1, 2, 4, 5, 10, 20
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36
Here greatest and common factor of 20 and 36 is 4.
There fore HCF of 20 and 36 is 4.
Least Common Mul t iple
The least common multiple of two or more given numbers is
the least of their common multiples.
eg, Multiple of 25 are 25, 50, 75, 100, 125, 150, 175,
....
Multiple of 30 are 30, 60, 90, 120, 150, 180, 210, ....
Here 150 is least common multiple of 25 and 30
There fore LCM of 25 and 30 is 150.
Using Division Method
Example 1: Determine the HCF and LCM of 36, 48,
64 and 72.
Solution. To find HCF
36 ) 48 ( 1
36
12 ) 36 ( 3
36
×
12 ) 64 ( 5
60
4 ) 12 ( 3
12
×
4 ) 72
(18
4
32
32
× There fore HCF of 36, 48, 64 and 72 is 4.
To find LCM
2
|
36 48
64 72
|
2
|
18 24
32 36
|
2
|
9 12 16
18
|
2
|
9 3
4
9
|
3
|
3 1
4
3
|
4
|
1 1
4 1
|
1 1
1 1
|
LCM = 2 × 2 × 2 × 2 × 3 × 4 = 576
HCF and LCM of Fractions
HCF of fraction = HCF
of Numerators
LCM of
Denominators
LCM of fraction = LCM of
Numerators
HCF of
Denominators
Example 2: Determine the HCF and LCM of 14
, 42 , 21 .
33 55 22
Solution. HCF of 14
, 42 ,21 .
33
55 22
= HCF of 14,42,21
LCMof 33,55,22
Now, to determine HCF of 14, 42 and 21
14 ) 42 ( 3
42
×
14 ) 21 ( 1
14
7 ) 14 ( 2
14
×
There fore HCF of 14, 42, 21 is 7
And to determine LCM of 33, 52, 22
11
|
33 55
22
|
3 5
2
|
There fore LCM of 33, 55 and 22 = 11 × 3 × 5 ×
2 = 330
Hence, Required HCF = 7
330
Now, LCM of 14
, 42 , 21
33 55 22
= LCMof 14,42,21
HCF of 33,55,22
To determine LCM of 14, 42, 21.
7
|
14 42
21
|
2
|
2 6
3
|
3
|
1 3
3
|
1 1
1
|
There for LCM of 14, 42, 21 = 7 × 2 × 3 = 42
To determine HCF of 33, 55, 22
33 ) 55 ( 1
33
22 ) 33 ( 1
22
11 ) 22 ( 2
22
×
11 ) 22 ( 2
22
×
There fore HCF of 33, 55, 22 = 11
Hence, Required LCM = 42
11
• The least number which is exactly divisible by a,
b and c is the LCM of a, b, c.
• The greatest number that will divide a, b,
c is the HCF of a, b, c.
• If x is a factor of a and b, then x
is also a factor of a + b, a – b and ab.
• HCF of given numbers must be a factor of their LCM.
• The product of the LCM and HCF of two numbers is equal
to the product of the two numbers.
Example 3: LCM of two numbers is 56 times of
their HCF. Sum of LCM and HCF is 456. If one of them is 56.
Find the other.
Solution. Let HCF be x, then LCM = 56x
x + 56x = 456 Þ x = 8
There fore HCF = 8 and LCM = 448
Now, 8 × 448 = 56 × other number
So, other number = 8 x 448
56
= 64
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