Teachers Eligibility Test - Paper 2
Class – VI Mathematics
NUMBER SYSTEM
Prime Factorization:
Expressing a given number as a product of
factors that are all prime numbers is called the prime factorization of
a number.
For example, 36 can be written as product of factors as
36 =1 x 36;
36 = 2 x 18;
36 = 3 x 12;
36 = 4 x 9;
36 = 6 x 6
Here, the factors of 36 can be found easily
as 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Note that not all the factors of 36 are prime
numbers.
There are two prime factorization methods.
They are
1. Division Method
2. Factor Tree Method
1. Division Method:
Find the prime factorization of 60.
60
= 2x2x3x5
Hence
the factors are 2x2x3x5.
2. Factor Tree Method:
Find the prime factorisation of 60.
60
= 2 x 30 = 2 x 15 = 3 x 5
Hence
the factors are 2x2x3x5.
Common Factors
Consider the numbers 45 and 60.
Use of divisibility tests will also help us to find the factors of 45 and 60.
The
factors of 45 are 1,3,5,9,15 and 45 and the factors of 60 are
1,2,3,4,5,6,10,12,15,20, 30 and 60.
Here, the common factors of 45 and 60 are 1,
3, 5 and 15.
Highest Common Factor (HCF)
The Highest Common Factor HCF
of two or more numbers is the highest number that divides the numbers exactly.
Example
Find the HCF of 63 and 42.
Solution:
The
prime factorisation of 63 = 3 x 3 x 7
The
prime factorisation of 42 = 2 x 3 x 7.
Then the common prime factors of 63 and 42 are 3 and 7
And so the highest common factor is 3 x 7 =21.
|
·
The Highest Common Factor (HCF) is also called as the
Greatest Common Divisor (GCD) or the Greatest Common Factor (GCF). ·
HCF (1, x) =1 ·
HCF (x, y) = x, if y is a multiple of x.
For example, HCF (3, 6) = 3. ·
If the HCF of two numbers is 1, then the numbers are
said to be co-primes or relatively prime. Here, the two numbers can both be primes as (5, 7) or both
can be composites as (14, 27) or one can be a prime and other a composite as
(11, 12). |
Example 3:
Find the HCF of the numbers 24 and 48 by
division method.
Solution:
The prime factorisation of 24 = 2x2x2x3
The prime factorisation of 48 =
2x2x2x2x3
Here, the prime factorisation of 24 = 2x2x2x3
The prime factorisation of 48 =
2x2x2x2x3
The product of common factors of 24 and 48 = 2 x 2 x 2 x
3 = 24
And so, HCF (24, 48) = 24.
Common Multiples:
Let us now write the
multiples of 4 and 10.
Multiples of are 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,…
Multiples of 10 are
10,20,30,40,50,60,70,80,90,100,…
Here, the common
multiples of 4 and 10 are 40
and 60 and will go on
without ending.
As multiples of a number
are infinite, we can think of the Least Common Multiple of numbers,
shortly denoted as LCM.
Least Common Multiple (LCM):
We can find the least
common multiple of two or more numbers by the following methods.
1.
Division Method
2. Prime Factorisation Method
Example:
Find the LCM of 144 and
198.
By Division Method
LCM = product of all prime factors
= 2 x 3 x 3 x 8 x 11
= 1584
Thus, the LCM of 144 and 198 is 1584.
By Prime Factorisation Method
144 = 2 x 72
= 2 x 3 x 24
= 2 x 3 x 3
x 8
198 = 2 x 99
= 2 x 3 x 33
= 2 x 3 x 3
x 11
Now, LCM = (product of common factors) X (product of factors that are
not common)
= (2 x 3 x 3) x (8
x 11)
= 18 x 88
= 1584
Thus, LCM of 144 and 198 is 1584.
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