 ## 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, M10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …}
The multiples of 12, M12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The multiples of 15, M15 = {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, M12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The multiples of 16, M16 = {16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, …}
The multiples of 24, M24 = {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

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

Solution: Here,
Multiples of 30, M30 = {30, 60, 90, 120, 150, 180, 210, 240, …}
Multiples of 40, M40 = {40, 80, 120, 160, 200, 240, 280, …}
Multiples of 60, M60 = {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.