This is second part of Factors and Multiples discussion. For to see our first discussion Please click on the link below.

###

###

###

###

###

:. LCM of 4 & 6 = 2x2x3 = 12

:. LCM of 6, 8, 12, 16 = 2x2x2x3x2 = 48

The prime factorization may be used to calculate LCM also. Let us try to apply it to find out the LCM of 6, 8, 12, 16.

6 = 2x3

8 = 2x2x2

12 = 2x2x3

16 = 2x2x2x2

To calculate the LCM we take the maximum occurrence of each prime factor. For example, 2 is a prime factor. Its maximum occurrence is 4 times (in 16). the maximum time 3 occurs in any of the factors is 1 ( in 12 & 6). So, the LCM = 2

Also multiples of 48 are divisible by 6, 8, 12 ,16.

i.e. 96, 144, 192, 240 ... are divisibel by 6, 8, 12, 16.

But since 48 is the smallest among all the common multiples, it is the LCM.

If P is divisible by Q, it is also divisible by the factors of Q i.e. if 90 is divisible by 30, 90 is divisible by factors of 30 (1, 2, 3, 5, 6, 10, 15)

Similarly, If P is a multiple of Q, any number R, that is a multiple of Q. For Example, 12 is a multiple of 6. So, 24, 36, 48 etc. Which are multiples of 12 are also multiples of 6.

Example:

LCM of 12 & 20 = 60

GCD of 12 & 20 = 4

Here, product of the two numbers = 12 x 20 = 240

LCM x GCD = 60 x 4 = 240

:. LCM x GCD = Product of the two numbers.

For to see our first discussion please click on the link below:

### Factors, Multiples, GCF & LCM : Factors, Prime Factorization, Factorization

###
**Multiples: **Multiples of a number are those which are divisible by the number.

### Multiples of 2: 2, 4, 6, 8, 10, 12 .............

### Multiples of 3: 3, 6, 9, 12, 15 .................

### A number has an infinite number of multiples. But any number has a fixed & finite number of factors.

###
**LCM: **Least common Multiple(LCM). A multiple is a always greater than or equal to the number white a factor is always less than or equal to the number.

### Multiples of 4: 4, 8, 12, 16, 20, 24,28 ............

### Multiples of 6: 6, 12, 18, 24, 30 .............

### Common multiples of 4 & 6 are 12, 24, 36, ....

### The least is 12.

### :. LCM of 4 & 6 is 12

### It can be shown by Venn Diagram also.

###
**Simplest way to find out LCM:**

**Example:**Find LCM of 6, 8, 12, 16.:. LCM of 6, 8, 12, 16 = 2x2x2x3x2 = 48

The prime factorization may be used to calculate LCM also. Let us try to apply it to find out the LCM of 6, 8, 12, 16.

6 = 2x3

8 = 2x2x2

12 = 2x2x3

16 = 2x2x2x2

To calculate the LCM we take the maximum occurrence of each prime factor. For example, 2 is a prime factor. Its maximum occurrence is 4 times (in 16). the maximum time 3 occurs in any of the factors is 1 ( in 12 & 6). So, the LCM = 2

^{4}x 3 = 48.**Note:**48 is divisible by 6, 8, 12, 16.Also multiples of 48 are divisible by 6, 8, 12 ,16.

i.e. 96, 144, 192, 240 ... are divisibel by 6, 8, 12, 16.

But since 48 is the smallest among all the common multiples, it is the LCM.

If P is divisible by Q, it is also divisible by the factors of Q i.e. if 90 is divisible by 30, 90 is divisible by factors of 30 (1, 2, 3, 5, 6, 10, 15)

Similarly, If P is a multiple of Q, any number R, that is a multiple of Q. For Example, 12 is a multiple of 6. So, 24, 36, 48 etc. Which are multiples of 12 are also multiples of 6.

**Once Useful Formula:**LCM x GCD = Product of the two numbers.Example:

LCM of 12 & 20 = 60

GCD of 12 & 20 = 4

Here, product of the two numbers = 12 x 20 = 240

LCM x GCD = 60 x 4 = 240

:. LCM x GCD = Product of the two numbers.

For to see our first discussion please click on the link below: