**L.C.M. by Long Division Method**:

A least common multiple of two numbers is the smallest positive number that is a multiple of both.

Multiple of 3 — 3, 6, 9, 12, 15, 18,…………..

Multiple of 4 — 4, 8, 12, 16, 20, 24,………….

So the LCM of 3 and 4 is 12, which is the lowest common multiple of 3 and 4.

**An example of LCM**

The LCM of 10, 20,25 is 100. It means that 100 is the lowest common multiple of these three numbers, but there is a question in our mind that can the LCM be (-100)? Since (-100) is lower than 100 and divisible by each of 10, 20, 25, or can it be zero or what will be the LCM of (-10) and 20? Will it be (-20) or (-200)?

For all these questions, there is only one answer that the LCM is only defined for positive numbers and LCM is not defined for 0.

**PROCESS OF FINDING LCM**

- We will do prime factorization in first step of all the numbers.
- Then we calculate the number of times each prime occurs in prime factorization and write each number as power of primes.
- Then in last step we write all the primes involved and raise each of the primes to highest power present.

**Example based on above **

**Example 1:LCM of 10, 20, 25?**

**Step 1**: 10= 2 × 5

20 = 2 × 2 × 5

25= 5 × 5

**Step 2****:** 10= 2^{1} × 5^{1}

20 = 2^{2} × 5^{1}

25= 5^{2}

**Step 3****:**Primes involved are 2 and 5

Now we raise each of the primes to highest power present i.e.2^{2}^{ }× 5^{2} =100. So 100 is required LCM.

**Example 2: What is the LCM of 35, 45, 55?**

**Step-1: **

35 = 5 × 7

45 = 3 × 3 × 5

55 = 11 × 5

**Step-2: **35 = 5

^{1}× 7

^{1}

45 = 3

^{2}× 5

^{1}

55 = 11

^{1}× 5

^{1}

**Step-3: Primes involved are 3, 5, 7 and 11**

Now we raise each of the primes to the highest power present i.e. 3

^{2}× 5

^{1}× 7

^{1}× 11

^{1}

LCM of the given numbers = 3465