LCM Formula

How is LCM Calculated?

In mathematics computation of the least common multiple and greatest common divisors of two or more numbers. LCM formula is defined as the smallest integer which is a multiple of two or more numbers. For example, LCM of 4 and 6 is 12,  and LCM of 10 and 15 is 30. As with the greatest common divisors, there are many methods for computing the least common multiple formulae. One method is the method to factor both numbers into their primes. The LCM is the product of all primes that are common to all numbers. In this topic, we will discuss what is LCM method, formula to find lcm, and LCM formula with examples. Let us learn it!

LCM by Listing Multiples -

Step 1: You need to list the multiples of each number until at least one of the multiples appears on all the lists.

Step 2: Now find the smallest number that is on all of the lists

Step 3: This number is the Least Common Multiple.

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Example: Let’s Find the LCM of 42 and 8.

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Example: Let’s Find the LCM of (6,7,21)

Write down the multiples of 6:  6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Write down the multiples of 7:  7, 14, 21, 28, 35, 42, 56, 63

Write down the multiples of 21: 21, 42, 63

Now you need to find the smallest number that is present on all of the lists. So the LCM(6, 7, 21) is 42.

To Find Out LCM Using Prime Factorization Method:

Step 1: You need to show each number as a product of their prime factors.

Step 2: LCM will be the product of the highest powers of all prime factors.

Now we know what is LCM method, let’s solve a few examples.

For Example, Let’s Find the LCM (12,30) We Find:

First find the prime factorization of 12 = 2 × 2 × 3

Second, the prime factorization of 30 = 2 × 3 × 5

Using all prime numbers we have found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60

Therefore, the LCM (12,30) = 60.

To Find Out the LCM Using Division Method:

Step 1: First, we need to write the given numbers in a horizontal line separated by commas.

Step 2: Then, we need to divide all the given numbers by the smallest prime number.

Step 3: We now need to write the quotients and undivided numbers in a new line below the previous one.

Step 4: Now you need to repeat this process until we find a stage where no prime factor is common.

Step 5: LCM will be the product of all the divisors and the numbers in the last line.

Now we know what is lcm method, let’s solve a few examples.

For Example, Let’s Find the LCM (12,30) We Find:

First find the prime factorization of 12 = 2 × 2 × 3

Second, the prime factorization of 30 = 2 × 3 × 5

Using all prime numbers we have found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60

Therefore, the LCM (12,30) = 60.

Least Common Multiple Formula for Any Two Numbers:

1) For two given numbers if we know their greatest common divisor that is GCD, then the Least Common Multiple can be calculated easily using the help of given least common multiple formula:

2) To get the LCM of two Fractions, then first we need to compute the LCM of Numerators and HCF of the Denominators. Further, both these results will be expressed as a fraction. Thus,

Questions to be Solved :

Question 1: Find out the LCM of 8 and 14.

Solution: 

Step 1: First you need to  write down each number as a product of prime factors.

8 = 2× 2 × 2 = 2³

14 = 2 × 7

Step 2: Product of highest powers of all prime factors.

Here the prime factors are 2 and 7

The highest power of 2 here = 2³

The highest power of 7 here = 7

Hence LCM = 2³ × 7 = 56

Question 2: Find the LCM(12,18,30).

Solution: List down the prime factors of 12 = 2 × 2 × 3 = 22 × 31

List down the prime factors of 18 = 2 × 3 × 3 = 21 × 32

List down the prime factors of 30 = 2 × 3 × 5 = 21 × 31 × 51

You need to list all the prime numbers found, the number of times as they occur most often for any one given number and you need to multiply them together to find the Least common multiple.

After multiplying, 2 × 2 × 3 × 3 × 5 = 180

Using the concept of exponents instead, multiply together each of the prime numbers with the highest power

In exponential form, 22 × 32 × 51 = 180

So, the LCM(12,18,30) = 180