In arithmetic, mathematics, and number system, the least common multiple ( LCM ) is also known as the smallest common multiple or lowest common multiple ( LCM ) of two integers. For any two integers p and q, usually denoted by LCM(p, q), is the smallest positive integer that is divisible by both p and q.
The LCM is the Lowest Common Denominator (LCD), that can be used before fractions can be added, subtracted, or compared. The LCM of more than two integers (like for 3 or 4 or 5 any times ) is also well-defined since LCM is the smallest positive integer that is divisible by each of them.
What is the Least Common Multiple(LCM)?
It is a method to find the smallest common multiple between numbers ( any two or more). Generally according to definitions the common multiple is a number which is a multiple of two or more numbers therefore LCM is used to determine the least common factor or multiple ( Generally called as Tables) of an integer. Since the division of integers by zero is undefined or not possible, this definition has meaning only if a and b are both different from zero (so not zero ).
Example 1:-
If we consider an example, To find L.C.M of 2 and 3?
Now multiples of 2 = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 , 24...
Again Multiples of 3 can be written as = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27,30, 33. 36, 39.42. 45, 48. ...
Common Multiples = 0, 6, 9, 18, 24, ....
Therefore least among common Factors ( LCM) = 6 ( where 6 is the smallest common multiple for numbers 2 and 3.)
Algorithm to find LCM of Numbers:-
- By Finding the Multiples,
- By Prime Factorization
- By Multiple Tree Diagram.
-
The method to find the least common multiple of any given numbers is first to write down the multiples (Tables) of individual numbers and then find the first common multiple between them.
Example 2:- To find LCM of 3 & 4 ? Solution:- Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24 ... Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, …. Common Multiples = 12, 24, ....
Therefore least among common Factors ( LCM) = 12 ( where 12 is the smallest common multiple for numbers 3 and 4. ) Example 3:-To find LCM of 3 4 & 6 ( Three numbers)?
Solution:-
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24 , 27,30, 33... Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36,40, 44, 48,…. Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54... Common Multiples = 12, 24,36 ....
Therefore least among common Factors ( LCM) = 12 ( where 12 is the smallest common multiple for numbers 3, 4 and 6.)
2 By Prime Factorization
Most Popular method to find the LCM of the given numbers is by prime factorization.
Example 4:-To find LCM of 6 16 & 20 ( Three numbers)?Solution :- There are given three numbers 6, 16, and 20. Now we have to write the prime factors of all three numbers individually.
6 = 2 x 3 == 2 x 3 x 1 ( Also written as )
16 = 2 x 2 x 2 x 2 = 2 x 2 x 2 x 2 x 1 ( Also written as )
20 = 2 x 2 x 5 = 2 x 2 x 5 x 1 ( Also written as )
On pairing the common prime factors of above numbers we get the LCM. Hence, there are four pairs of 2 and one pair of 3 and 5 each. So the LCM of 6, 16 and 20will be;
LCM (6, 16, 20) = 2 x 2 x 2 x 2 x 3 x 5 = 240 3 By Multiple Tree Diagram.
Example 5:- To find LCM of 6 & 10?
The multiple trees can be formed by using the prime factorization method this is similar to the above method.
Therefor LCM of 6 and 10 = 1 x 2 x 3 x 5 = 30.
this method is known as Tree Diagram.
Use's of LCM:-
LCM can be used before the addition, subtraction, or comparison of fractions can take place, Therefore it is basic fundamental tools for using mathematical operations in Fractions.
The LCM of more than two integers also happens to be well-defined: it is the smallest positive integer whose division can take place by each of them.
________________________________________________
Posts
