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, and 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.
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
| Properties | Description |
|---|---|
| Associative property | LCM(a, b) = LCM(b, a) |
| Commutative property | LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c)) |
| Distributive property | LCM(da, db, dc) = dLCM(a, b, c) |
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 number 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 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 the 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.
| Prime factors | 1st number | 2nd number | 3rd number |
|---|---|---|---|
| 2 | 10 | 18 | 20 |
| 2 | 5 | 9 | 10 |
| 3 | 5 | 9 | 5 |
| 3 | 5 | 3 | 5 |
| 5 | 5 | 1 | 5 |
| 1 | 1 | 1 |
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, that 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 the LCM of two numbers here.
LCM Table
| Numbers | LCM |
|---|---|
| 24 and 36 | 72 |
| 10 and 15 | 30 |
| 8 and 10 | 40 |
| 15 and 20 | 60 |
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, that 12 is the smallest common multiple.
Therefore, LCM. of 4, 6, and 12 is 12.
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.
