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Least Common Multiple (L.C.M.)

The abbreviation L.C.M. stands for Least Common Multiple. The lowest possible number that can be divisible by both numbers is the least common multiple of two numbers. We can calculate two or more integers as well as two or more fractions with the help of L.C.M. 

There are various methods to find the L.C.M. of 2 numbers, the easiest way to find the L.C.M. of two numbers is to use the prime factorization of each number and then the product of the highest powers of the common prime factors be the L.C.M. of those numbers.

Now let’s understand the actual Definition of L.C.M.?

The Least common multiple is also called the Lowest common multiple. Amongst all the given numbers, the least common multiple of two or more numbers is the smallest number among all common multiples. 

Now the question comes: How can we find the L.C.M.?

L.C.M. of numbers can be found in multiple ways but there are 3 basic methods to find the least common multiple of any two numbers. Let’s learn the methods now:

  • Listing Method
  • Prime Factorization
  • Division Method

Let’s start with the first method: Listing Method:

By this method, we can find out the common multiples of the two or more numbers.  Amongst these common multiples, the least common multiple is considered and the LCM of 2 given numbers can then be calculated. How can we calculate the L.C.M. of the two numbers P and Q by the listing method?

Follow the steps given below:

  • List the first few multiples of P and Q.
  •  Note down the common multiples of both numbers. 
  • Select the smallest common multiple. That smallest common multiple is called as 

the lowest common multiple or the L.C.M. of the two numbers. 

Prime Factorisation Method:

By this method we can find out the prime factors of the numbers and these prime factors can be used to find the L.C.M. of those numbers. 

Follow the steps given below:

  • List down the numbers in the prime factored form. 
  • The L.C.M. of the mentioned two numbers is the product of all the prime factors. (But the common factors will be included only once)

Division Method:

By using this method we can divide the numbers by a common prime number and then  these common prime factors are used to calculate the L.C.M. of the mentioned numbers. 

Follow the steps given below:

  • Step 1: Find a prime number which is a factor of at least one of the mentioned numbers. Write down that prime number on their left hand side of the mentioned number. 
  • Step 2: If the prime in step one could be an issue of the amount, then divide the amount by the prime and write the quotient below. If the prime in step one isn’t an element of the amount, then write the amount within the row below because it is. press on with the steps until one is left within the last row.

We saw all the three methods but amongst all these three methods mentioned above, the Division Method is the easiest and time-saviour method.

Now let’s learn some Formulas of L.C.M.:

For Integers:

Let’s suppose ‘a’ and ‘b’ are the two integers, then the formula goes like:

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

For Fractions:

Let’s suppose a/b and c/d are the two fractions, then the formula goes like:

LCM (a/b, c/d) = (LCM of Numerators)/(HCF of Denominators) = LCM (a,c)/HCF (b,d)

Relation between Highest Common Factor and Lowest Common Factor:

H.C.F. of 2 or more numbers is the Highest Common Factor of the given numbers. Whereas the least common multiple of 2 or more numbers is the smallest number among all common multiples of the given numbers. Let’s assume ‘a’ and ‘b’ are the two numbers then the formula that gives the relationship between their L.C.M. and H.C.F. is given as:

LCM (a,b) × HCF (a,b) = a × b

Differentiating L.C.M. and H.C.F.:

LCM (Lowest Common Multiple)HCF (Highest Common Factor)
The least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers.The highest common factor of two or more numbers is the highest number among all the common factors of the given numbers.
The respective numbers are the factors of their LCM.HCF of two or more numbers is a factor of each of the numbers.
LCM of two or more prime numbers is always the product of those numbers.HCF of two or more prime numbers is 1 always.
LCM of two or more numbers is always greater than or equal to each of the numbers.HCF of two or more numbers is always less than or equal to each of the numbers.

Understanding The Properties of L.C.M.:

The L.C.M. of 2 or more numbers is having a wide range of its properties but as of now let’s focus on the three major properties which are as follows:

  • Associative Properties Of L.C.M.: This property is used for 2 numbers- 

LCM (a,b) = LCM (b, a).

  • Commutative Properties Of L.C.M.: This property is used for 3 numbers-

 LCM (a,b,c) = LCM (a, LCM (b,c)) = LCM (LCM (a,b),c) = LCM (LCM (a,c),b).

  • Distributive Properties Of L.C.M.: This property is only used for 4 numbers-

 LCM (da, db, dc) = d × LCM (a, b, c).

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