**Prime Numbers**

**Prime Numbers**

A whole
number greater than 1 that has only two factors, 1 and itself, is called a
prime number.

For example,
2, 3, 5, 7, 11, 13, 17, 19, 23, etc. are all prime numbers.

##
**Composite
Numbers**

**Composite
Numbers**

A whole
number greater than 1 that has more than two factors is called a composite
number.

For example,
4, 6, 8, 9, 10, 12, 14, 15, etc. are all composite numbers.

** Co-primes **

A pair
of two natural numbers having no common factor other than 1 are called

**co-primes**. The numbers may or may not be primes.
For
example, (2, 3), (3, 5), (6, 35), (9, 16), (5, 12), etc.

## **Twin Primes **

Pairs of
prime numbers that differ by 2 are called

**twin primes**.
For
example, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), etc.

** ****Prime Triplets **

The set of three consecutive
prime numbers, i.e. (3, 5, 7), (11, 13, 17), etc. are called

**prime triplets**.##
##
**Prime
Factorization**

##
**Prime
Factorization**

A prime
number that is a factor of a composite number is called a prime factor of the
composite number.

The process
to express a composite number as a product of prime factors only is called
prime factorization.

There are
two methods two find the prime factorization of a composite number.

1. Factor Tree Method

2. Continuous Division Method

###
###
**Factor
Tree Method**

###
**Factor
Tree Method**

**Example 1:**Find the prime factorization of 340.

**Solution:**We can construct a factor tree as follows.

Step 1: Write
down two factors whose product is 340 as follows.

Step 2: Continue
to factorize any factors which is a composite number.

Step 3: Stop
this process when the last row of the tree shows only prime factors. The product
of all the prime factors in the tree yields the prime factorization of the
given number.

Therefore, 340 = 2 × 2 × 5 × 17

###
###
**Continuous
Division Method**

###
**Continuous
Division Method**

We can also
express 340 as a product of its prime factors using the continuous division
method by dividing 340 by the smallest prime numbers such as 2, 3, 5, etc.

Therefore, 340
= 2 × 2 × 5 × 17

##
##
**Highest
Common Factor (HCF)**

##
**Highest
Common Factor (HCF)**

The highest
common factor (HCF) of a group of numbers is the largest number that can divide
all the numbers in the group.

For example,
let us find the HCF of 18 and 24.

The factors
of 18:

**1, 2, 3,****6,**9, 18
The factors
of 24:

**1, 2, 3**, 4,**6,**8, 12, 24
We can see
that 1, 2, 3, and 6 are the common factors of 18 and 24. The largest of these
common factors is 6. Thus, 6 is the highest common factor (HCF) of 18 and 24.

There are
two methods to find the HCF of the given numbers.

1. Prime Factorization Method

2. Common Division Method

###
###
**Prime
Factorization Method**

###
**Prime
Factorization Method**

**Example 2:**Find the HCF of 750 and 225 using prime factorization.

**Solution:**We find the prime factorization of each number from any one of the given methods.

750 = 2 × 3 × 5 ×
5 × 5

225 = 3 × 3 × 5 ×
5

HCF = 3 × 5 × 5

= 75

###
###
**Common
Division Method**

###
**Common
Division Method**

**Example 3:**Find the HCF of 750 and 225 using common division method.

**Solution:**In this method, we start dividing by smallest prime factor of both the numbers and continue until both the numbers does not have any common prime factor. The product of all the common prime factors is the required HCF.

HCF = 3 × 5 × 5 = 75

Therefore, the HCF of 750 and 225 is 75.

##
##
**Least
Common Multiple (LCM)**

##
**Least
Common Multiple (LCM)**

The least
common multiple (LCM) of a group of numbers is the smallest number which is
divisible by all the numbers in the group.

For example,
let us find the LCM of 6 and 8.

The
multiples of 6: 6, 12, 18,

**24**, 30, 36, 42,**48**, 54, 60, …
The
multiples of 8: 8, 16,

**24**, 32, 40,**48**, 56, 64, 72, 80, …
We can see
that 24 and 48 are the first two common multiples of 6 and 8. Since 24 is the
least of all the common multiples, we say that the least common multiple (LCM)
of 6 and 8 is 24.

There are
two methods to find the LCM of the given numbers.

1. Prime Factorization Method

2. Long Division Method

###
###
**Prime
Factorization Method**

###
**Prime
Factorization Method**

**Example 4:**Find the LCM of 20, 24, and 70 using prime factorization method.

**Solution:**We find the prime factorization of each number from any one of the given methods.

20 = 2 × 2 × 5

24 = 2 × 2 × 2 × 3

70 = 2 × 5 × 7

LCM = 2 × 2 × 2 × 3 × 5 × 7

= 840

Therefore, the LCM of 20, 24 and 70 is 840.

###
###
**Long
Division Method**

###
**Long
Division Method**

**Example 5:**Find the LCM of 24 and 90 using long division method.

**Solution:**In this method, we start dividing by smallest prime factor of one or both the numbers and continued dividing until you get 1. The product of all the divisors is the required LCM.

LCM = 2 × 2 × 2 × 3 × 3 × 5

= 360

Therefore, the LCM of 24 and 90 is 360.

##
##
**Relationship between HCF and LCM of
Two Numbers **

##
**Relationship between HCF and LCM of
Two Numbers **

Let
us consider any two prime numbers. Their HCF is always 1, and LCM is equal to
their product. For example, 5 and 7 are two prime numbers.

∴ LCM of 5
and 7 = 5 × 7 = 35

HCF of 5 and 7 = 1

Product of given
numbers = 5 × 7 = 35

Also, product of HCF
and LCM = 1 × 35 = 35

So, HCF × LCM = Product
of given prime numbers

Let us consider any two
composite numbers, say, 12 and 16.

HCF of 12 and 16 = 4

LCM of 12 and 16 = 48

Now, product of HCF and
LCM = 4 × 48 = 192

Product of numbers = 12
× 16 = 192

∴

**HCF × LCM = Product of given composite numbers**

**Example 6:**The LCM of 16 and another number is 48. If their HCF is 8, find the other number.

**Solution:**Product of given numbers = LCM × HCF

∴ 16 × other number = 48 × 8

∴ other number = (48 ×
8)/16 = 24

So, the other number is 24.

**Example 7:**The LCM and HCF of 60 and another number are 240 and 20 respectively. Find the other number.

**Solution:**Product of given numbers = LCM × HCF

∴ 60 × other number = 240 × 20

∴ other number = (240
× 20)/60 = 80

So, the other number is 80.

**Example 8:**The HCF of 25 and 70 is 5. Find their LCM.

**Solution:**Product of given numbers = LCM × HCF

∴ 25 × 70 = LCM × 5

∴ LCM = (25 × 70)/5 = 350

So, the LCM is 350.Study Divisibility Tests in detail ---- Click Here!

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