Least common multiple of two numbers is the smallest number which is a multiple of both the given numbers.
For example, consider the whole numbers 2 and 3.
Now, list out the whole numbers which are multiples of 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 smallest whole number which is a multiple of both 2 and 3.
In other words, 6 is the smallest 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
We can find the least common multiple of two or more numbers by the following methods.
1. Division Method
2. Prime factorization Method
(LCM ----> Least Common Multiple)
Example :
Find the LCM of 156 and 124.
Solution by Division Method :
Step 1 :
Start with the smallest prime factor and go on dividing till all the numbers are divided as shown below.
Step 2 :
LCM = Product of all prime factors
= 2 x 2 x 3 x 13 x 31
= 4836
Thus, the LCM of 156 and 124 is 4836.
Solution by Prime Factorization Method :
Step 1 :
Write the prime factors of 156 and 124 as shown below (use of divisibility test rules will also help).
156 = 2 x 2 x 3 x 13
124 = 2 x 2 x 31
Step 2 :
The prime factor 2 appears a maximum of 2 times in the prime factorization of 156 and 124.
The prime factors 3 and 13 appear only 1 time in the prime factorization of 156, the prime factor 31 appears only 1 time in the prime factorization of 124.
Hence, the required LCM is
= (2 x 2) x 3 x 13 x 31
= 4836
Problem 1 :
Find the LCM of the following set of numbers using prime factorization method.
6, 9
Solution :
Step 1 :
Write the prime factors of 6 and 9 as shown below.
6 = 2 x 3
9 = 3 x 3
Step 2 :
The prime factor 3 appears a maximum of 2 times in the prime factorization of 9.
The prime factor 2 appears only 1 time in the prime factorization of 6.
Hence, the required LCM is
= (3 x 3) x 2
= 9 x 2
= 18
Problem 2 :
Find the LCM of the following set of numbers using prime factorization method.
8, 12
Solution :
Step 1 :
Write the prime factors of 8 and 12 as shown below.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
Step 2 :
The prime factor 2 appears a maximum of 3 times in the prime factorization of 8.
The prime factor 3 appears only 1 time in the prime factorization of 12.
Hence, the required LCM is
= (2 x 2 x 2) x 3
= 8 x 3
= 24
Problem 3 :
Find the LCM of the following set of numbers using division method.
16, 24
Solution :
Step 1 :
Start with the smallest prime factor and go on dividing till all the numbers are divided as shown below.
Step 2 :
LCM = Product of all prime factors
= 2 x 2 x 2 x 2 x 3
= 48
Thus, the LCM of 16 and 24 is 48.
Problem 4 :
Find the LCM of the following set of numbers using division method.
64, 80
Solution :
Step 1 :
Start with the smallest prime factor and go on dividing till all the numbers are divided as shown below.
Step 2 :
LCM = Product of all prime factors
= 2 x 2 x 3 x 2 x 7 x 5
= 840
Thus, the LCM of 64 and 80 is 840.
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