How to Find LCM?

There are 3 methods to find the least common multiple of two numbers.

• LCM by Listing Method
• LCM by Prime Factorization Method
• LCM using Division Method

LCM by Listing Method

We can find out the common multiples of two or more numbers by listing their multiples. And out of these common multiples and the least common multiple is considered to be the LCM of two given numbers.

Follow the steps below to calculate the LCM of the two numbers A and B by the listing method:

Step 1: List down the first few multiples of A and B.

Step 2: Mark the common multiples from the multiples of both numbers.

Step 3: Select the smallest marked common multiple. Hence, this results in the LCM(A, B).

Example: Find LCM of two positive integers 2 and 6.

Answer:

• Multiples of 2: 2,4,6,8,10,12,14…
• Multiples of 6: 6,12,18,24, 30…

The common multipliers of 2 and 6 are 6, 12…, So, the least common multiple is 6.
Hence, LCM(2, 6) = 6

LCM by Prime Factorization Method

We can find LCM using the Prime factorization method of the given numbers. 

Follow the steps below to calculate the LCM of two numbers using the prime factorization method:

1. First, find the prime factors of the given numbers using the repeated division method.
2. Write these numbers in the form of an exponent and find the product of only those prime factors that have the highest power.
3. The product of these factors with the highest powers is the LCM of the given numbers

Example: Find LCM of two positive integers 120 and 300.

Answer:

• The prime factorization of 120 are: 2*2*2*3*5 = 2³*3¹*5¹
• The prime factorization of 300 are: 2*2*3*5*5 = 2²*3¹*5²

Now, find the product of only those factors that have the highest powers among these. This will be, 2³ * 3¹ * 5² = 8 * 3 *  25 = 600

Hence, LCM(120, 300) = 600

LCM by Division Method

We can find LCM using the Division method of the given numbers. This can be done by dividing the numbers by a common prime number, and these prime factors are used to calculate the LCM of those numbers. 

Follow the steps below to calculate the LCM of the two numbers A and B by Division Method:

Step 1: Find a prime number which is a factor of at least one of the given numbers. Write this prime number on the left of the given numbers.

Step 2: If the prime number in Step 1 is a factor of the number, then divide the number by the prime and write the quotient below it. If the prime number in step 1 is not a factor of the number, then write the number in the row below as it is. Continue the steps until 1 is left in the last row.

Example: Let’s take two positive integers 3 and 4, the task is to find the LCM(3, 4).

Answer:

LCM(3,4) = 12

The LCM is the product of all these prime numbers.
Hence, LCM(3, 4) = 12

HCF (Highest Common Factor) and LCM (Least Common Multiple) concepts are the foundation of many mathematical operations and are essential in solving complex problems. HCF and LCM problems challenge your ability to find the greatest common factor and the smallest common multiple of numbers, and they require both logical and mathematical skills. So get ready to exercise your brain as we delve into the world of HCF and LCM problems and explore the exciting ways they can be used to solve challenging aptitude questions!