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Multiple of a number is obtained by multiplying that number by a natural number. A number has infinitely many multiples. Two or more numbers have so many multiples that are common. The smallest of these common multiples is called the Lowest common multiple or L.C.M.

### Lowest Common Multiple (L.C.M.)

LCM of 12 and 16 using the prime factorization method.

The lowest common multiple of two or more numbers is the smallest number which is a multiple of each of the numbers.

The lowest common multiple of two or more numbers is the smallest number which is divisible by all the given numbers. This means that there cannot be a number divisible by the given numbers and smaller than the lowest common multiple.

For example. consider numbers 8 and 12.

Multiples of 8 are: 8, 16, **24**, 32, 40, **48**, 56, 64, **72….**

Multiples of 12 are: 12, **24**, 36, **48**, 80,** 72**, …

Common multiples are: 24,48,72. …

Clearly, 24 is the smallest among common multiples.

Therefore, L.C.M. of 8 and 12 is **24.**

There are two methods to find the L.C.M. of given numbers:

**(1)** **Prime factorization method**

**(2)** **Common division method**

### Prime Factorization Method

In order to find the LC.M. of two or more numbers by prime factorization method, we follow the following steps:

**STEP 1) **Obtain the numbers,

**STEP 2)** Write prime factorization of each of the numbers and express them in exponential form.

**STEP 3)** Find the product of all different prime factors with highest power in the prime factorization of each number.

**STEP 4)** The number obtained in Step (3) is the required L.C.M.

We illustrate the above method by means of the following examples.

#### Illustrative Example 1

Find the LC.M. of 36 and 60 by prime factorization method.

**Solution:**

**STEP 1)** Obtain the numbers 36 and 60

**STEP 2)** Express each number as a product of prime factors

Factors of 36 = 2 x 2 x **3 x 3**

Factors of 60 = **2 x 2** x 3 x **5**

In these prime factorizations the maximum number of times, the prime factor 2 occurs is two; this happens for 36 and 60. Similarly, the maximum number of times

the prime factor 3 occurs is two; this happens for 36. The prime factor 5 occur only one in 60.

Therefore, Required L.C.M. = (2 x 2) x( 3 x 3) x 5= **180**.

#### Illustrative Example 2

Find the LC.M. of 112, 168 and 266 by prime factorization method.

**Solution:**

**STEP 1)** Obtain the numbers 112, 168 and 266

**STEP 2)** Express each number as a product of prime factors

Factors of 112 = **2 x 2 x 2 x 2** x 7

Factors of 168 = 2 x 2 x 2 x **3** x** 7**

Factors of 266 = 2 x 7 x **19**

In these prime factorizations the maximum number of times, the prime factor 2 occurs is four; this happens for 112. Similarly, the maximum number of times

the prime factor 3 occurs is one; this happens for 168. The prime factor 7 occur only one in 112 and 168. Similarly, the prime factor 19 occur only one in 266.

Therefore, Required L.C.M. = (2 x 2 x 2 x 2) x 3 x 7 x 19= **6384**.

What is the Least Common Multiple (LCM) 12, and 18? Here we will show you step-by-step how to find the Least Common Multiple of 4, 12, and 18.

### Common Division Method

We follow the following steps to find the L.C.M of given numbers by common division method:

**STEP 1)** Arrange the given numbers in a row separated by commas.

**STEP 2)** Obtain a number which divides exactly at least two of the given numbers.

**STEP 3)** Divide the numbers which are divisible by the number chosen in Step (2) and write the quotients just below them. Carry forward the numbers which are not divisible.

**STEP 4)** Repeat Step (2) and (3) till no numbers have a common factor other than **1**.

**STEP 5)** LCM is the product of the divisors and the remaining numbers.

We illustrate the above method by means of the following example.

365/838 is already in the simplest form.