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FINDING LOWEST COMMON MULTIPLE (LCM)

__FINDING LOWEST COMMON MULTIPLE (LCM) -__

**The Least Common Multiple (LCM) is a mathematical concept used to find the smallest multiple that two or more numbers have in common. In other words, it's the smallest positive integer that is evenly divisible by all the given numbers.**

**Finding the LCM involves identifying the prime factors of the numbers and then determining how many times each prime factor needs to be included in the LCM. Here's a step-by-step process to find the LCM of two or more numbers:**

__Prime Factorization:-__Express each number as a product of its prime factors. For example, if you have the numbers 12 and 18: 12 = 2^2 * 3^1 or 12 = 2² X 3¹ (prime factors: 2 and 3) 18 = 2^1 * 3^2 or 18 = 3² X 2¹ (prime factors: 2 and 3)__Identify Common Factors:-__List all the distinct prime factors that appear in the prime factorization of the given numbers. In this case, the common prime factors are 2 and 3.__Highest Powers:-__For each common prime factor, select the highest power it appears with in any of the prime factorization. In this case:Highest power of**2: 2^2****or****2²****Highest power of****3: 3^2****or****3²**__Multiply:-__Multiply the highest powers of the common prime factors to get the LCM: LCM = 2^2 * 3^2 = 4 * 9 = 36 or 2² X 3²= 4 X 9 = 36.

**So, the LCM of 12 and 18 is 36.**

**If you need to find the LCM of more than two numbers, follow the same steps by considering all the prime factors involved.**

**LCM is used in various mathematical and real-world applications, such as in solving equations, working with fractions, and determining patterns in periodic events or cycles.**

**The Least Common Multiple (LCM) is a mathematical concept used to find the smallest multiple that two or more numbers have in common. In other words, it's the smallest number that is divisible by each of the given numbers without leaving a remainder.**

**To find the LCM of two or more numbers, you can use several methods. Here's a common approach:**

__Method 1:-__ Prime Factorization

**Start by finding the prime factorization of each of the numbers.****Identify all the distinct prime factors involved.****For each prime factor, take the highest power that appears in any of the numbers' factorizations.****Multiply all the prime factors raised to their respective highest powers to get the LCM.**

__Method 2:-__ Listing Multiples

**List the multiples of each number until you find a common multiple.****The smallest common multiple is the LCM.**

__Method 3:-__** Division**

**Divide the product of the given****numbers by their greatest common divisor (GCD).****The result of this division is the LCM.**

**Let's go through an example using the first method:**

**Example.1) Find the LCM of 36 and 72.**

**Prime factorization of 18: 18 = 2^1 * 3^2 or 18 = 3² X 2¹****Prime factorization of 36: 12 = 2^2 * 3^2 or 36 = 3² X 2²****The distinct prime factors are 2 and 3. Take the highest power for each: 2^2 and 3^2.****Multiply these highest powers: LCM = 3^2 * 2^2 = 9 * 4 = 36.**

**So, the LCM of 18 and 36 is 36. (Ans.)**

**1) Finding LCM By Prime Factorization Method,**

**2) Finding LCM By Repeated Division Method,**