Least common multiple is the least value among the multiples of the given numbers.
For example, consider the numbers 2 and 3.
Now, list out the whole numbers which are multiples of both both 2 and 3.
6, 12, 18, 24, ..........
All the above numbers are multiples of both 2 and 3.
Is there any whole number less than 6 which is a multiple of both 2 and 3?
The answer is NO.
So, 6 is the least whole number which is a multiple of both 2 and 3.
In other words, 6 is the least whole number which is evenly divisible by both 2 and 3.
Therefore, 6 is the least common multiple of 2 and 3.
More Examples :
Least common multiple of 4 and 6 = 12
Least common multiple of 10 and 15 = 30
Least common multiple of 5 and 10 = 10
The above examples of finding least common multiple of two whole numbers are pretty easy. Incase, we have bogger numbers or more than two numbers, how to find the least common multiple?
We can find the least common multiple of three numbers by the following methods.
1. Division Method
2. Prime factorization Method
(LCM ----> Least Common Multiple)
Example 1 :
Find the least common multiple of the following set of numbers using division method.
30, 40, 60
Solution :
Start with the smallest prime factor and go on dividing till all the numbers are divided as shown below.
LCM = Product of all prime factors
= 2 x 5 x 3 x 2 x 2
= 120
Thus, the least common multiple of 30, 40 and 60 is 120.
Example 2 :
Find the least common multiple of the following set of numbers using prime factorization method.
27, 45, 81
Solution :
Write the prime factors of 27, 45 and 81 as shown below.
27 = 3 x 3 x 3
45 = 3 x 3 x 5
81 = 3 x 3 x 3 x 3 x 3
The prime factor 3 appears a maximum of 5 times in the prime factorization of 81.
The prime factor 5 appears only 1 time in the prime factorization of 45.
Hence, the required LCM is
= (3 x 3 x 3 x 3 x 3) x 5
= 81 x 5
= 405
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