HCF AND LCM OF NUMBERS
Factor – The numbers which are
multiplied together are called the factors of the product.
e.g. – 70 =
1,
2, 5, 7, 10, 14 and 35 are factors of 70.
24 =
1, 2, 3, 4, 6, 8, 12
and 24 are factors of 24.
Finding factors of a number –
We can find the
factors of a product by two method –
(A)Using Multiplication –
In this method, we try
to find the numbers whose product is the given number.
E.g. – let the number
= 24
Since
= 24. Therefore, 1, 24 are factors of 24.
Since
Therefore, 2, 12 are factors of 24.
Since
.
Therefore, 3, 8 are factors of 24.
Since
Therefore, 4, 8 are factors of 24.
Therefore, all the factors
of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
(B)Using Division -
In this method, we try
to find a number which divides the given number without leaving any remainder.
e.g. – let the number = 24
Since
Since
Since
Since
Therefore,
all the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Properties of factors –
a)
1
is a factor of every number.
We know that,
Thus, 1 is a factor of
every number.
b)
Every
counting number is a factor of itself.
We know that,
Thus, every counting
number is a factor of itself.
c)
Every
counting number is a factor of 0.
We know that,
Thus, every counting
number divides zero (0).
d)
A
factor of a counting number is either less than or equal to that number.
It is clear that,
The smallest
factor of a counting number is 1.
The largest factor of a counting number is
the number itself.
So, the factor of a
counting number is either less than or equal to the number.
e)
Division
by 0 is meaningless.
Multiples – The product of two or
more numbers is called the multiple of each of those numbers.
e.g. – multiples of 4 are
=
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44,
Properties of multiples –
a)
Every
number is a multiple of 1.
We know that,
Thus, every number is a
multiple of 1.
b)
Every
counting number is a multiple of itself.
We know that,
Thus, counting number
is a multiple of itself.
c)
0
is a multiple of every counting number.
We know that,
Thus, zero (0) is a
multiple of every counting number.
d)
Every
multiple of a counting number is either equal to or greater than the number.
We know that,
The smallest multiple of 4 is 4; the smallest
multiple of 9 is 9; the smallest multiple of
20 is 20
so on.
Thus, every
multiple of a counting number is either greater than or equal to the number.
Prime factors – Factors of a number which are prime are called its prime
factors.
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24
Prime factors of 24 are: 2, 3
Factorization – Expressing a number as the product of its factors is
called factorization.
Prime factorization – A factorization in which every
factor is prime is called prime factorization of the number.
e.g.-
|
5
|
25
|
|
|
5
|
||
When a number is written as a product of its prime factors,
it is said to be completely factorized.
Factor Tree – Prime factorization of a number can also
be found by factorizing in a pictorial form, called a factor
tree.
|
2
|
|
54
|
|
27
|
|
2
|
|
3
|
|
9
|
|
3
|
|
3
|
Common Factors
– A
number is said to be a common factor of two or more numbers, if it is a factor
of each of them.
e.g.-
The
factors of 12 = 1, 2, 3, 4, 6, 12.
The
factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.
The
factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24.
Thus,
the common factors of 12, 36 and 24 are 1, 2, 3, 4 and 6.
Co-Prime Numbers
– A
pair of numbers which do not have a common factor other than 1, are called
co-primes.
e.g.-
3,
5; 9, 16 are co-prime numbers.
The
factors of 12 = 1, 2, 3, 4, 6, 12.
The
factors of 25 = 1, 5, 25.
Thus, 12 and 25 have only one common i.e.
1.
Therefore,
12 and 25 are co-prime numbers.
Highest Common Factor (HCF)
Or, Greatest Common Divisor (GCD)
Or, Greatest Common Measure (GCM) –
The common factor of two or more numbers having
the highest value among all the factors is called the Highest Common Factor of
those numbers.
e.g.-
Factors
of 20 are 1, 2, 4, 5, 10 and 20.
Factors
of 35 are 1, 5, 7 and 35.
Common
factors of 20 and 35 are 1 & 5.
Out
of these, 5 is greater than 1.
Thus,
the HCF of 20 and 35 is 5.
There are three methods which are
commonly used to find the HCF of two or more numbers –
1.
H.C.F. by Prime
Factorization Method
2.
H.C.F. by
Common Division Method
3.
H.C.F. by
Continuous Division Method
Prime
Factorization Method –
e.g. – Find the HCF of 72 and 90
by prime factorization method.
|
|
|
|
2
|
72
|
|
2
|
36
|
|
2
|
18
|
|
3
|
9
|
|
3
|
3
|
|
|
1
|
Prime
factorizations of 72 and 90 are:
72
=
|
2
|
90
|
|
3
|
45
|
|
3
|
15
|
|
5
|
5
|
|
|
1
|
90 =
Thus, HCF of 72
and 90 is
Common Division
Method –
e.g. – Find the HCF of 90, 126
and 270 by common division method.
|
2
|
90, 126,
270
|
|
3
|
45, 63,
135
|
|
3
|
15, 21,
45
|
|
|
5,
7, 15
|
HCF
=
Continuous
Division Method –
e.g. – Find the HCF of 144, 408
and 468.
144) 408 ( 2
-288
120) 144 (1
-120
24) 120 (5
-120
0 ∴
HCF of 144 and 408 is 24.
Now, we find the HCF of 24 and
468.
24 )
468 ( 19
-24
228
-216
12 ) 24 ( 2
-24
0
Thus, the HCF of
144, 408, and 468 is 12.
Common
Multiples – A
number is said to be a common multiple of two or more numbers, if it is a
multiple of each of them.
Or, If a number is a multiple of
two or more numbers, it is called a common multiple of the numbers.
e.g.
–
The
multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, … … … … … … … … …
The
multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, … … … … … … … … …
The
multiples of 6 = 6, 12, 18, 24, 30, 36, 42, … … … … … … … …
Common
multiples of 2, 3 & 6 are 6, 12, 18, … … … … … …
Lowest Common
Multiples (LCM) –
The common multiple of
two or more numbers having the least value among all the multiples is called the common multiple
of those numbers.
Or, The lowest common multiple of two or more numbers is the smallest out of
all their common multiples.
e.g. –
Find the LCM of 18 and 54.
Multiples of 18 are 18, 36, 54, 72, 90, 108, 162,
144, 162, 180, … … … … … … … … …
Multiples of 54 are 54, 108, 162, 216, 270, … … … … …
… …
Common multiples of 18 and 54 are 54, 108, 162, … … …
… …
Thus, LCM of 18 and 54 is 54.
We can find the LCM of two or more
numbers by the following three methods –
1.
L.C.M. by Prime
Factorization Method
2.
L.C.M. by
Common Divisor Method
3.
L.C.M. with the
help of the HCF
Prime
Factorization Method -
|
2
|
84
|
|
2
|
42
|
|
3
|
21
|
|
7
|
7
|
|
|
1
|
e.g.
– Find the LCM of 60 and 84 by prime factorization method.
|
2
|
60
|
|
2
|
30
|
|
3
|
15
|
|
5
|
5
|
|
|
1
|
Thus,
the LCM of 60 and 84 is 420.
Common Division
Method –
e.g. – Find the
LCM of 12, 15, 18 and 20 by common division method.
|
2
|
12, 15,
18, 20
|
|
2
|
6,
15, 9, 10
|
|
3
|
3,
15, 3, 5
|
|
5
|
1,
5, 3, 5
|
|
|
1,
1, 3, 1
|
Thus,
the LCM of 12, 15, 18 and 120 is 180.
With the Help
of the HCF –
Find
the LCM of 868, 1922 and 2108.
868
) 2108 ( 2 124
) 1922 ( 15
-1736 -1860
372 ) 868 ( 2 62 ) 124 ( 15
-744 -124
124 ) 372 ( 3 0
-372
0 ∴ HCF = 62
|
62
|
868, 1922,
2108
|
|
2
|
14,
31, 34
|
|
|
7,
31, 17
|
Thus, LCM
Relationship
between two numbers and their HCF and LCM –
|
|
For the HCF of 12 & 18. For
the LCM of 12 & 18.
|
2
|
12, 18
|
|
3
|
6,
9
|
|
|
2,
3
|
12
) 18 ( 1
-12
6 ) 12 ( 2
12
0 LCM
=
The HCF of 12 & 18 = 6 The LCM of 12 & 18 = 36
The product of
HCF and LCM =
The product of
12 and 18 =
(HCF)
(LCM) = (1st number)
(2nd number)
HCF
=
LCM =
1st
number =
2nd
number =
No comments:
Post a Comment