LCM of Two Numbers

The Least Common Multiple is the smallest of the common multiples. To understand the concept of least common multiple, we need to know what multiples and common multiples mean.

Multiple: When we multiply a number, let’s say 5 by another number such as 4 (never with a ‘0’), the result that we get is 20. i.e., 20 is the multiple of 5. It is just like the multiplication table. 

For example: the multiples of 4 are: 4,8,12,16,20,24,28,32,36,40 and so on....

Common Multiple: From the word ‘common’, we know that it means similarities between two or more things. Let’s say we have listed the first few multiples of 4 and 5. Thus, the common multiples are those found in both lists. Given below are the multiples of 4 and 5 ( common multiples highlighted in yellow):

Notice that numbers 20 and 40 are common in both lists.

So, we can say that the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)

Least Common Multiple (LCM): It is the smallest positive number that is a common multiple of two or more numbers. In other words, a common multiple can be defined as a number that is a multiple of two or more numbers. 

Example 1: The multiples of 3 and 5 are:

Multiples of 3 - 3, 6, 9, 12, 15, 18,....

Multiples of 5 - 5, 10, 15, 20, ....

The least common multiple of 3 and 5 is 15 because it is the smallest number which is common in both the tables.

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How to find LCM?

There are various ways to find the least common multiple (LCM) of two numbers:

1. Writing down the multiples and finding the smallest common multiple.

2. Calculating LCM from the prime factors of the given numbers.

3. Division Method or Ladder Method.

Method 1: Writing down the multiples and finding the smallest common multiple: One of the easiest ways to find the least common multiple of two numbers is to first list the prime factors of each number.

Example 2: The common multiples of 3 and 4 are 0, 12, 24,...

The least common multiple (LCM)  is the smallest common number which is found in the multiples of both 3 and 4; hence, we can say 12 and 24(excluding 0) are the multiples of both 3 and 4.   

Method 2:  Calculating LCM from the prime factors of the given numbers: 

The most common way of finding LCM is the prime factorization method. 

Example 3: Find the LCM of 30 and 45 by using the prime factorisation method.

Solution: To find the LCM of two numbers say, 30 and 45, the steps are as follows:

The first step is to find the LCM of 30 and 45,

30 = 2 × 3 × 5

45 = 3 × 3 × 5

Then, the second and final step is to multiply each factor by the greatest number of times it appears in either number. If the same factor appears more than once in both numbers, you multiply the factor to the greatest number of times it appears. Here is an example.

2: one occurrence

3: two occurrences

5: one occurrence

2 × 3 × 3 × 5 = 90 <— LCM

Method 3: Division Method or Ladder Method of Finding LCM:

In this method, the two numbers are divided simultaneously with prime numbers until the division is even. When no more primes are left that evenly divide into both numbers, multiply the divisors to get the LCM. 

Example 4: Find the LCM of 2940 and 3150

First, you need to factor in each of the numbers. 

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Start by dividing 2940 by the smallest prime that would fit into it, being 2. You will be left with another even number, 1470. 

Divide it again by 2. The result, 735, is divisible by 5, but it is also divisible by 3. And to a matter of fact, 3 is a smaller number between both of them so we divide the result by 3 to get 245. Now, 245 cannot be divided by 3 but by 5. So we divide it by 5 and the outcome will be 49, which is divisible by 7.

The same sequential-division process will be applied to 3150: 

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Here too, each of the given numbers was divided by the smallest primes that fit into them, until the result would be a prime result. So the prime factorizations of 2940 and 3150 are:

2940 = 2 × 2 × 3 × 5 × 7 × 7

3150 = 2 × 3 × 3 × 5 × 5 × 7

Write down all the factors neatly and according to their occurrence. 

2940: 2 × 2 × 3 × 5 × 7 × 7

3150: 2 × 3 × 3 × 5 × 5 × 7

The procedure will become much easier if each of the factors will be in its column. 

The Least Common Multiple, the LCM, is the smallest ("least") number of all the common multiples of  2940 and 3150. That is, it is the smallest number that contains both 2940 and 3150 as factors. The smallest number that is a multiple of both of these values; is the common multiple to the two values. Therefore, it will be the smallest number that contains every factor in these two numbers.

Looking back at the listing, we can see that 3150 has one copy of the factor of 2; while 2940 has two copies. Since the LCM must contain all factors of each number, the LCM must contain both copies of 2. However, to avoid overduplication, the LCM does not need three copies because neither 2940 nor 3150 contains three copies.

So, the LCM of 2940 and 3150 must contain both copies of factor 2. By the same reasoning, the LCM must contain both copies of 3, both copies of 5, and both copies of 7:

2940: 2×2×3  ×5  ×7×7

3150: 2  ×3×3×5×5×7

LCM: 2×2×3×3×5×5×7×7 = 44,100

Thus, the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.

Difference between GCF & LCM

LCM is the least common multiple of any set of numbers whereas GCD or GCF is the highest divisor of the numbers. 

There are differences between GCF and LCM which are as follows:

Did You know?

• The concept of LCM was first introduced by mathematician Euclid

• LCM is used for encoding messages and coding and Cryptography

• LCM is also used to optimise the quantity that is under operations 

Sample Questions 

1. Find the LCM of 15 & 25 using the ladder method.

Ans: Using the ladder method, we can find the LCM of 15 and 25 as follows

The first step is to consider the common multiple of both the numbers and then divide.

The second step is to write its factors below the number

 As

5 ||   15    25

   ||     3      5

No number will divide 3 and 5 simultaneously.  and hence the LCM of 15 and 25 is number 5.

2. Find the LCM of the following set of numbers: 6,18,14,12.

Ans: For finding the LCM of 6,18,14,12, the prime factorization method will be used.

Step 1: the number should be written as a multiple of their prime factors. 

and hence the numbers can be written as

6 = 2 x 3

18 = 2 x 3 x 3

14 = 2 x 7

12 = 2 x 2 x 3

Step 2: Determine the common integer that is present in all the given numbers.

We can see that too is the only member that appears in all the numbers.  whereas the mite uh the other number such as 3 and 7 might appear in some but not in other and hence the LCM of the numbers 6,18, 14, and 12 is 2

3. Find the LCM of ¾ and ⅚. 

Ans: The LCM of the fraction that is the formula to be followed is as follows LCM of any fraction is equal to the LCM of the numerator divided by the HCF of the denominator. 

Here in both the fractions, numerators are 3 & 5. And denominators are 4 & 6. 

So LCM of numerators will be 3 x 5 = 15. (As both the numbers 3 & 5 have nothing in common)

And HCF of denominators, 4 & 6 will be 12.

As 4 = 2² and 6 = 2 x 3

So the HCF will be 2² x 3 = 12. 

Therefore LCM of ¾ and ⅚ is 15/12.

And the fraction 15/12 can further be reduced to 5/4. 

Hence, LCM of ¾ and ⅚ is 5/4.