##
**LCM
(Lowest Common Multiple)**

LCM (Lowest Common Multiple) of given two or more than
two numbers is a lowest one number which is exactly divisible by the given
numbers. Among the common multiples of the given numbers, the lowest one is the
LCM.

For example, 60 is the lowest number which is exactly
divisible by 10, 12 and 15. So 60 is the LCM of 10, 12 and 15.

Here,

The multiples
of 10,

**M**_{10}= {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …}
The multiples
of 12,

**M**_{12}= {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The multiples
of 15,

**M**_{15}= {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, …}
Common
multiples =

**{60, 120, …}**
The lowest
common multiple =

**60****∴**

**LCM of 10, 12 and 15 is = 60**

###
**LCM
can be calculated by using the following methods:**

**a.**

**Set of multiple method**

**b.**

**Prime factorization method**

**c.**

**Common division method**

Lets find LCM of 12, 16 and 24 by three different
methods.

####
**Set
of multiple method**

The
multiples of 12,

**M**_{12}= {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The
multiples of 16,

**M**_{16}= {16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, …}
The
multiples of 24,

**M**_{24}= {24, 48, 72, 96, 120, 144, 168, 192, 216, 240, …}
Common
multiples =

**{48, 96, 144, …}**
The
lowest common multiple =

**48****∴**

**LCM = 48**

####
**Prime
factorization method**

To find LCM by prime factorization method, we should
apply the following steps:

Step I: Find prime factors
of the given numbers and list them.

Step II: Write the common
factors and the remaining factors.

Step III: Find the product of
these factors.

Here,

**∴**

**12 = 2×2×3**

**16 = 2×2×2×2**

**24 = 2×2×2×3**

The product of the common and remaining factors is the LCM.

**∴**

**LCM = 2 × 2 × 2 × 3 × 2 = 48**

####
**Common
division method**

To calculate LCM by common division method, we should
apply the following steps:

Step I: Arrange the given
numbers in a row and separate by commas.

Step II: Divide all numbers
by the smallest prime numbers.

Step III: Repeat the process
till the numbers left in the last row is 1.

Step IV: Multiply all the
prime divisors to get LCM.

Let’s find the LCM of 12, 16 and 24 by common division
method.

**∴**

**LCM = 2 × 2 × 2 × 2 × 3 = 48**

###
*Workout
Examples*

*Workout Examples*

*Example 1: Find the LCM of 30, 40 and 60 by the set of multiples method.*

*Solution:*

*Here,*

*Multiples of 30, M*_{30}= {30, 60, 90, 120, 150, 180, 210, 240, …}

*Multiples of 40, M*_{40}= {40, 80, 120, 160, 200, 240, 280, …}

*Multiples of 60, M*_{60}= {60, 120, 180, 240, 300, 360, …}

*Common multiples = {120, 240, …}*

*Lowest common multiple is the LCM,*

*∴*

*LCM = 120*

*Example 2: Find the LCM of 50, 60 and 80 by the prime factorisation method.*

*Solution:*

*The prime factors of 50, 60 and 80 are calculated as follows:*

*∴*

*50 = 2×5×5*

*60 = 2×2×3×5*

*80 = 2×2×2×2×5*

*∴*

*LCM = Common factors × remaining factors*

*= 2 × 2 × 5 × 5 × 3 × 2*

*= 1200*

*Example 3: Find the LCM of 30, 40 and 80 by using common division method.*

*Solution:*

*Here,*

*∴*

*LCM = 2 × 2 × 2× 2 × 3 × 5 = 240*

*Example 4: Find the least number which is exactly divisible by 12, 40 and 60 without leaving a remainder.*

*Solution:*

*The required least number is the LCM of 12, 40 and 60.*

*∴*

*LCM = 2 × 2 × 2× 3 × 5 = 120*

*∴*

*The required least number is 120.*

*Example 5: Find the least number which when divided by 40, 60 and 80 leaves a remainder 6 in each case.*

*Solution:*

*Here,*

*∴*

*LCM = 2 × 2 × 2× 2 × 3 × 5 = 240*

*∴*

*The required number = LCM + remainder*

*= 240 + 6*

*= 246*

*You can comment your questions or problems regarding the LCM here.*

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