The least common multiple or LCM or LCD. There are two methods to find lcm.
A) Lcm prime factorization
B) By using multiples.
The lowest common multiple of two or more given numbers is the lowest (or smallest or least) of their common multiples. By individual prime factors
1) LCM prime factorization : In this method use prime numbers find the factors of the given numbers individually. Multiplication of all the prime numbers gives you the LCM. Example 1: Find the LCM of 12 and 18 Solution:The prime factorization of 12 and 18 are :
12 = 2 x 2 x 3
18 = 2 x 3 x 3
In these LCM prime factorization, the maximum number of times the prime factor 2 occurs is two; this happens for 12. Similarly , the maximum number of times the factor 3 occurs is two; this happens for 18. The LCM of the two numbers is the product of prime factors counted the maximum number of times they occur in any of the numbers. Thus in this case
LCM = 2 x 2 x 3 x 3 = 36 Example 2 : Find the LCM of 24 and 90. Solution : The prime factorization of 24 and 90 are :
24 = 2 x 2 x 2 x 3
90 = 2 x 3 x3 x 5
In these prime factorizations the maximum number of times the prime factor 2 occurs is three; this is for 24. And the maximum number of times the prime factor 3 occurs is two; this is for 90. The prime factor 5 occurs only once in 90 .
Thus LCM of 24 and 90 is
= (2 x 2 x 2) x ( 3 x 3 ) x 5 = 360.
By common factorization (LCM Prime Factorization)
By common prime factorization Example 1 : Find the LCM of 20,25 and 30. Solution :
1) Divide by the least prime number which divides at least one of the given numbers. Here 2 is the least prime number. But the number 25 is not divisible by 2 so write 25 as in it is in the next row.
2) Continue this till we have no multiples of 2.
3) Then take the next prime number 3.Continue till we have no multiples of 3.
4) Divide by next prime number 5 till there is no multiple of 5.
Thus, LCM of 20,25 and 30 is
= 2 x 2 x 3 x 5 x 5 = 300 B) By using multiples
In this method write the tables of the given numbers. The least common number gives you the LCM. This method is used for small numbers only. Example : Find the LCM of 8 and 12.
Multiples of 8 = 8,16,24,32,40,48,56,64,72,80,…
Multiples of 12 = 12, 24,36,48, 60,72,84,96,108,120,…
Here the common multiples are 24,48,72, …
But the least common multiple is 24.
Thus, the LCM of 8 and 12 is 24.