LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

LEAST COMMON MULTIPLE (LCM)

__LEAST COMMON MULTIPLE (LCM)__

**Any
number can be a multiple of two or more numbers. Suppose there are two numbers
these are 2 and 4.**

**To
find any number of common multiple of the given numbers we should have to find
the first common multiple and then multiply it 1 time, 2 times, 3 times.**

**Product
of 2 & 4 is = 2 X 4 = 8.**

**Thus,
the common multiples of 2 & 4 are – ****8 x 1 = 8**

** 8 x 2 = 16**

** 8 x 3 = 24**

** 8 x 4 = 32**

** 8 x 5 = 40, …………**

**
So, 8 is the LCM of 2 & 4.**

**Write the 5 common multiple of the following numbers**

**Example.1) 3
& 7**

**Ans.)
Product of 3 & 7 is = 3 X 7 = 21**

**5
common multiplies of the following multiplies are = 21 x 1 = 21**

**
21 x 2 = 42**

**
21 x 3 = 63**

**
21 x 4 = 84**

**
21 x 5 = 105**

**So,
LCM of the number 3 & 7 is 21 and first five multiples are 21, 42, 63 ,
84, 105.**

**Example.2) 5 & 6**

**Ans.)
Product of 5 & 6 is = 5 X 6 = 30**

**5
common multiples of the following multiplies are = ****30 x 1 = 30**

**
30 x 2 = 60**

**
30 x 3 = 90**

**
30 x 4 = 120**

**
30 x 5 = 150.**

**So,
LCM of the number 5 & 6 is 30 and the first five multiples are 30, 60,
30, 120, 150.**

__DIVISIBILITY -__

**Now
we will learn, how to understand which ‘Dividend’ is divisible by which ‘
Divisor’.**

**Suppose
there is a number, that is 453 and we have to find which number will be
‘Divisor’,**

**Firstly
add all the digit of 453 like = 4 + 5 + 3 = 12 , 12 is divisible by 3**

__BASIC
FACTORIZATION -__

**Basic
Factorization is nothing but a simplified multiplication form.**

**For
an example, basic factorization of 24 = 1 x 2 x 2 x 2 x 3.**

**basic
factorization of 36 = 1 x 2 x 2 x 3 x 3.**

**a) 1 has only one factor and that is the number(1) itself. It is called ‘Unique‘
number.**

**b)
2, 3, 5, 7, 11, 13 have only two factors, 1 and the number itself, these
numbers are called ‘ Prime’ Numbers.**

**c)
Numbers having three or more factor like 4, 6, 8, 10, 12, ……… are called ‘Composite**’ **Number**.

__LCM BY BASIC FACTORIZATION THROUGH DIVISION METHOD –__

**Example.1) Find the Basic Factors of the number 50 -**

__Step.1)__ First, consider 50 as dividend and consider the lowest number which is 2 which is count as ‘Divisor’. 50 ÷ 2 = 25. So, 25 is a ‘Quotient’.

__Step.2)__ Put 25 below the position of 50 and then try to divide. We can find 5 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 25 and we can get 5 as ‘Quotient’.

__Step.3)__ Put 5 below the position of 25 and then try to divide. We can find 5 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 5 and we can get 1 as ‘Quotient’.

** So, basic factorization of 50 is 2 x 5 x 5 x 1 (Ans.)**

**Example.2) Find the Basic Factors of the number 80 -**

__Step.1)__ First, consider 80 as dividend and we can find the lowest number which is 2 and 2 will be considered as ‘Divisor’. 80 ÷ 2 = 40. So, 40 is a ‘Quotient’.

__Step.2)__ Put 40 below the position of 80 and then try to divide. We can find 2 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 40 and we can get 20 as ‘Quotient’ .

__Step.3)__ Put 20 below the position of 40 and then try to divide. We can find 2 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 20 and we can get 10 as ‘Quotient’

__Step.4)__ Put 10 below the position of 20 and then try to divide. We can find 2 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 10 and we can get 5 as ‘Quotient’

__Step.5)__ Put 5 below the position of 10 and then try to divide. We can find 5 is the lowest ‘Divisor’ for dividing the ‘Dividend’ 5 and we can get 1 as ‘Quotient’.

** So, basic factorization of 80 is 2 x 2 x 2 x 2 x 5 x 1 (Ans.)**

**Example.3) Find the Basic Factors of the number 45 & 30 -**

__Step.1)__ First, consider 45 & 30 together as ‘Dividend’ and we can find the lowest number which is 5, and 5 is the common ‘Divisor’ of both the number 45 & 30. If 5 is ‘Divisor’ then, 45 ÷ 5 = 9 is ‘Quotient’ & 30 ÷ 5 = 6 is ‘Quotient’.

__Step.2)__ Now put 9 & 6 below the number 45 & 30 respectively. Consider 9 & 6 as ‘Dividend’ and find the lowest common ‘Divisor’ for both the number 9 & 6. We can find, 3 is the lowest common ‘Divisor’ for dividing both the ‘Dividend’ 9 & 6, we can get 9 ÷ 3 = 3 & 6 ÷ 3 = 2, 3 & 2 as ‘Quotient’

__Step.3)__ Now put 3 & 2 below the number 9 & 6 respectively. Consider 3 & 2 as ‘Dividend’ and find the lowest common ‘Divisor’ for both the number 3 & 2. We can find, 1 is the only lowest common ‘Divisor’ for dividing both the ‘Dividend’ 9 & 6 we can get 3 ÷ 1 = 3 & 2 ÷ 1 = 2, 3 & 2 as ‘Quotient’

**So, LCM by basic factorization of 45 & 30 is 5 x 3 x 1 x 3 x 2 = 90.**

** (Ans.)**

**Example.4) Find the Basic Factors of the number 24 & 36 -**

**Step.1)**** First, consider 24 & 36 together as ‘Dividend’ and we can find the lowest number which is 2, and 2 is
common ‘Divisor’ of both the number 24 & 36. If 2 is ‘Divisor’ then,
24 ÷ 2 = 12 is ‘Quotient’ & 36 ÷ 2 = 18 is ‘Quotient’.**

**Step.2)**** Now put 12 & 18 below the
number 24 & 36 respectively. Consider 12 & 18 as ‘Dividend’ and
find the lowest common ‘Divisor’ for both the number 12 & 18.**

**We can find, 2 is the
lowest common ‘Divisor’ for dividing both the ‘Dividend’ 12 & 18. we
can get 12 ÷ 2 = 6 & 18 ÷ 2 = 9, so respectively 6 & 9 is
‘Quotient’**

__Step.3)__ Now put 6 & 9 below the
number 12 & 18 respectively. Consider 6 & 9 as ‘Dividend’ and find
the lowest common ‘Divisor’ for both the numbers 6 & 9. We can find, 3
is the only lowest common ‘Divisor’ for dividing both the ‘Dividend’ 6 &
9. we can get 6 ÷ 3 = 2 & 9 ÷ 3 = 3. so respectively 2
& 3 is ‘Quotient’

__Step.4)__ Now put 2 & 3 below the
number 6 & 9 respectively. Consider 2 & 3 as ‘Dividend’ and find
the lowest common ‘Divisor’ for both the number 2 & 3. We can find, 1
is the only lowest common ‘Divisor’ for dividing both the ‘Dividend’ 2 &
3. we can get 2 ÷ 1 = 2 & 3 ÷ 1 = 3. so, respectively 2 & 3
is ‘Quotient’

**So, LCM by basic
factorization of 24 & 36 is 2 x 2 x 3 x 1 x 2 x 3 = 72 (Ans.)**

__LCM by basic factorization -__

**Example.1) Find
LCM of 15 & 25.**

**So, 15 =
3 x 5
x 1**

** 25 = 5 x 5
x 1**

**In the above condition
for both, the number of common factor is 5 & 1.**

**Now to find out the
LCM of 15 & 25 multiply common factor 5 ,1 and uncommon factors 3 & 5.**

** So, product of
the factors are = 3 x 5 x 5 x 1 = 75.**

**75 is the LCM of the
15 & 25.** **(Ans.)**

**There are some other way of solution of about Lowest Common Multiple (LCM) is given below for your more better understanding -**

**(i) Common Multiple Method Or Listing Method,**

**(ii) Common Division Method,**

**(iii) Prime Factorization Method,**