Least Common Multiple (LCM)

In Mathematics, the LCM of any two is the value that is evenly divisible by the two given numbers. The full form of LCM is Least Common Multiple. It is also called the Least Common Divisor (LCD). For example, LCM (4, 5) = 20. Here, the LCM 20 is divisible by both 4 and 5 such that 4 and 5 are called the divisors of 20.

LCM is also used to add or subtract any two fractions when the denominators of the fractions are different. While performing any arithmetic operations such as addition, subtraction with fractions, LCM is used to make the denominators common. This process makes the simplification process easier.

Definition

Least Common Multiple(LCM) is a method to find the smallest common multiple between any two or more numbers. A common multiple is a number which is a multiple of two or more numbers.

LCM denotes the least common factor or multiple of any two or more given integers. For example, L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80, where 80 is the smallest common multiple for numbers 16 and 20.

Now, if we consider the multiples of 16 and 20, we get;

16 → 16, 32, 48, 64, 80,… 

20 → 20, 40, 60, 80,…, 

We can see that the first common multiple for both numbers is 80. This proves the method of LCM as correct.

Also check: LCM of two numbers 

What is HCF?

Along with the least common multiple, you must have heard about the highest common factor, (H.C.F.). HCF is used to derive the highest common factors of any two or more given integers. It is also called as Greatest Common Divisor (GCD). 

For example, the H.C.F. of 2,6,8 is 2, because all the three numbers can be divided with the factor 2, commonly. H.C.F. and L.C.M. both have equal importance in Maths.

Properties of LCM

How to Find LCM?

As we have already discussed, the least common multiple is the smallest common multiple for any two or more given numbers.

A multiple is a value we get when we multiply a number with another number. Like 4 is a multiple of 2, as we multiply 2 with 2, we get 4. Similarly, in the case of the maths table, you can see the multiples of a number when we multiply them from 1, 2, 3, 4, 5, 6 and so on but not with zero.

LCM Formula

Let a and b are two given integers. The formula to find the LCM of a & b is given by:

LCM (a,b) = (a x b)/GCD(a,b)

Where GCD (a,b) means Greatest Common Divisor or Highest Common Factor of a & b.

LCM Formula for Fractions

The formula to find the LCM of fractions is given by:

L.C.M. = L.C.M Of Numerator/H.C.F Of Denominator

Different Methods of LCM

There are three important methods by which we can find the LCM of two or more numbers. They are:

• Listing the Multiples
• Prime Factorisation Method
• Division Method

Let us learn here all three methods:

Listing the Multiples

The method to find the least common multiple of any given numbers is first to list down the multiples of specific numbers and then find the first common multiple between them. 

Suppose there are two numbers 11 and 33. Then by listing the multiples of 11 and 33, we get;

Multiples of 11 = 11, 22, 33, 44, 55, ….

Multiples of 33 = 33, 66, 99, ….

We can see, the first common multiple or the least common multiple of both the numbers is 33. Hence, the LCM (11, 33) = 33.

LCM By Prime Factorisation

Another method to find the LCM of the given numbers is prime factorization. Suppose, there are three numbers 12, 16 and 24. Let us write the prime factors of all three numbers individually.

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

24 = 2 x 2 x 2 x 3

Now writing the prime factors of all the three numbers together, we get;

12 x 16 x 24 = 2 x 2 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Now pairing the common prime factors we get the LCM. Hence, there are four 2’s and one 3. So the LCM of 12, 16 and 24 will be;

LCM (12, 16, 24) = 2 x 2 x 2 x 2 x 3 = 48

LCM By Division Method

Finding LCM of two numbers by division method is an easy method. Below are the steps to find the LCM by division method:

• First, write the numbers, separated by commas
• Now divide the numbers, by the smallest prime number. 
• If any number is not divisible, then write down that number and proceed further
• Keep on dividing the row of numbers by prime numbers, unless we get the results as 1 in the complete row
• Now LCM of the numbers will be equal to the product of all the prime numbers we obtained in the division method

Let us understand with the help of examples.

Example: Find LCM of 10, 18 and 20 by division method.

Solution: Let us draw a table to divide the numbers by prime factors.

Therefore, LCM (10, 18, 20) = 2 x 2 x 3 x 3 x 5 = 180

Now, if we have to find the common multiple of two or more numbers, then we have to write all the multiples for the given numbers. Say for example, if there are two numbers 4 and 6, then how to find the common multiple between them?

LCM of Two Numbers

Let us write multiples of 4 and 6 first,

4 : 4,8,12,16,20,24,28,…..

6: 6,12,18,24,30,36,42…..

From the above two expressions you can see, 4 and 6 have common multiples as 12 and 24. They may have more common multiple if we go beyond. Now, the smallest or least common multiple for 4 and 6 is 12. Therefore, 12 is the LCM of 4 and 6. Also, learn to find LCM of two numbers here.

LCM Table

LCM of Three Numbers

Now, let us take an example of 3 numbers.

Example: Find the LCM 4,6 and 12.

Solution: First write the common multiples of all three numbers.

Common Multiples of 4 : 4,8,12,16,20,24,28,…..

Common Multiples of 6: 6,12,18,24,30,36,42…..

Common Multiples of 12: 12,24,36,48,60,72,….

From the above-given multiples of 4, 6 and 12, you can see, 12 is the smallest common multiple.

Therefore, LCM. of 4, 6 and 12 is 12.

Least Common Multiple Tree

The Least common multiple trees can be formed by using the prime factorisation method. Suppose there are two numbers 60 and 282. Then, first let us write the prime factors of these two numbers, such as;

60 = 6 x 10 = 2 x 3 x 2 x 5

282 = 2 x 141 = 2 x 3 x 47

Now let us represent the above prime factorization using a tree.

From the above tree diagram, we can take the pair of common factors and unique factors from the branches of both the numbers and multiply them as a whole to get the LCM. Therefore,

LCM (60, 282) = 2 x 2 x 3 x 5 x 47 = 2820

Relation Between LCM and HCF

LCM and HCF are the two important methods in Maths. The LCM is used to find the least possible common multiples of two or more numbers whereas HCF is a method to find the highest possible common factor among the numbers. But both can be related by the formulas:

LCM(a,b) = a × b / GCF(a,b)

GCF(a,b) = a × b / LCM(a,b)

Where a and b are the two numbers.

Solved Examples

Example 1: Find L.C.M. of 10 and 20.

Solution:

We know, for given two integers a and b,

L.C.M. (a,b) =(a * b)/GCD(a,b)

Therefore, L.C.M. (10,20) = (10 * 20)/GCD(10, 20)

The greatest common divisor for 10 and 20 is 10.

Thus, L.C.M. (10,20) =200/GCD(10)

L.C.M. (10,20) = 20

Example 2: If HCF of two numbers,12 and 30, is 6. Then find their LCM. 

Solution: As we know, 

LCM (a,b) = a × b / GCF(a,b)  [By the formula]

Therefore, 

LCM (12,30) = (12 x 30)/6      [given]

LCM (12, 30) = 60

Example 3: What is the smallest number when divided by the digits 5, 7, 9 and 12 gives the same remainder of 3 in every case?

Answer: 

To find the least common multiple of the divisors and add the remainder.

LCM (5, 7, 9, 12) = 1260

The required number = 1260 + 3 = 1263

Example 4: What is the greatest 4-digit number that is divisible by exactly by 15, 21 and 28?

Answer: 

The greatest 4-digit number is 9999.

LCM (15, 21, 28) = 420

Divide 9999 by 420, 339 is the remainder.

The required number is 9999 – 339 = 9660.

Example 5: The traffic policemen at 3 varied positions on the ground blow a whistle succeeding every 42 seconds, 60 seconds and 78 seconds respectively. If all of them blow at the same time at 9:30 hours, find the time interval at which they whistle again together.

Answer: 

The time interval at which they whistle again together = least common multiple of the separate whistleblowing pattern.

LCM (42, 60, 78) = 2 * 3 * 7 * 10 * 13 = 5460

The policemen will blow the whistle again at the same time interval after 5460 seconds.

On conversion into hours and minutes, it is 1 hour 31 minutes.

In terms of time, it is 11:01:00 hours.

Example 6: What is the greatest number when divided by the numbers 6, 7 and 8 leaves behind a remainder number 3, however when divided by number 9 has no remainder?

Answer: 

LCM (6, 7, 8) = 168

The number is of arrangement 168m + 3.

168m + 3 must be divisible by the number 9.

The given number is divisible by the number 9 when the total of its digits is a multiple of number 9.

When m = 1, the number is 168 + 3 = 171, the total of the digits is number 9.

The required number is 171.

Example 7: 2 digits are in the ratio 2:3. The product of the least common multiple and highest common factors is 294, what are the numbers?

Answer: 

Assume the ratio that is common to be m.

Let the numbers 2m and 3m.

Product of assumed numbers = product of least common multiple and highest common factor

=> 2m * 3m = 294

=> m² = 49

=> m = 7

The numbers are 2m = 2 * 7 = 14 and 3m = 3 * 7 = 21.

The required numbers are 14 and 21.

Practice Questions

1. Find the LCM of 15 and 13.
2. Find the LCM of 10 and 12
3. Find the LCM of 5, 10, 15 and 30
4. Find the LCM of 10 and 20 by the listing method.
5. Find the LCM by prime factorisation of 8 and 20.
6. Find the LCM of 15 and 24 by the division method.

Application of LCM-Video Lesson

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