## Least Common MultipleIn arithmetic, It is used when we add, subtract, or compare the fractions. While we perform addition or subtraction of the fractions, we find the LCM of the denominators and then solve the fractions. The LCM of the denominators is known as ## Properties of LCM**Associative:**LCM (a, b) = LCM (b, a)**Commutative:**LCM (a, b, c) = LCM (LCM (a, b), c) = LCM (a, LCM (b, c))**Distributive:**LCM (ka, kb, kc) = kLCM (a, b, c)**The LCM is related to GCF:**
## How to Find LCMThere are three method to find the LCM, are as follows: - Using Table Method
- Using the Greatest Common Divisor (GCD) Method
- Using Prime Factorization Method
- Using Multiples of Numbers
## Using Table MethodIt is a simple method and works for any number of numbers. Follow the steps given below to find the LCM. - List all the given numbers horizontally in the table separated by a comma.
- Start dividing the number(s) by 2 if the numbers(s) are completely divisible. Write 2 at the top of the left column and write the result horizontally. Repeat this process until we do not get the prime number(s) as a result.
- When any number is not divisible by 2, choose the next largest prime number and start dividing the number(s) by that number. Write that number below the 2 and write the result horizontally. Repeat this step until we do not get 1's in the last row.
- To get the LCM, multiply all the numbers written in the left-most column.
Let's understand it through examples.
## Using the Greatest Common Divisor (GCD) MethodWe can also calculate the LCM by using the GCD. The formula for LCM using the GCD is:
## How to Find GCDFollow the steps given below to find the GCD: - Write all the factors of each number.
- Select the common factors.
- Select the largest number, as GCD.
Let's understand it through examples.
According to the formula that we have learned above: First, we find the GCD of 8 and 10. Factors of 8: 1, 2, 4, 8 Factors of 10: 1, 2, 5, 10 Common Factors: 1, 2 Greatest Common Divisor: 2
According to the formula that we have learned above: First, we find the GCD of 11 and 42. Factors of 11: 1, 11 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Common Factors: 1 Greatest Common Divisor: 1
According to the formula that we have learned above: First, we find the GCD of 64 and 112. Factors of 64: 1, 2, 4, 8, 16, 32, 64 Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112 Common Factors: 1, 2, 4, 7, 8, 16 Greatest Common Divisor: 16
## Using Prime Factorization MethodFollow the steps to find the LCM using the prime factorization method. - Write all the prime factors of the given numbers.
- Select all the prime numbers. As many as they occur most often for anyone given number.
- Multiply all the prime numbers together to get the LCM.
## Note: When we write the factors in the form of is called exponential form, and the process is known as factorization using exponents.Let's understand through examples.
Prime factor of 17: 17
Prime factor of 17: 17
Prime factor of 35: 5×7
Prime factor of 35: 5
Prime factor of 223: 223
Prime factor of 223: 223
Prime factor of 12: 2 × 2 × 3
Prime factor of 12: 2
## Using Multiples of NumbersIt is a very lengthy method, so it is not usually used. Follow the steps given below to find the LCM using multiples of numbers. - List all the multiple of each number until the first common multiple is found.
- Pick the smallest multiple that is common in all the given numbers.
Let's understand it through examples.
LCM (15,16)=240
## LCM of FractionsWe can also find the LCM of the fractions by using the following formula:
Factors of 2: 1, 2 HCF: 2 Putting the values in the formula, we get:
Factors of 2: 1, 2 Factors of 3: 1, 3 Factors of 4: 1, 2, 4 Factors of 5: 1, 5 HCF: Putting the values in the formula, we get:
Factors of 9: 1, 3, 9 Factor of 1: 1 Factors of 5: 1, 5 HCF: Putting the values in the formula, we get:
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