LCM by Prime Factorisation – Mindspark

LCM- Definition, and Methods to calculate:

LCM or the Least Common Multiple of two or more numbers is the smallest number that is entirely divisible by all the given numbers. For example, if y is the LCM of a, b and c, then y is the smallest number which is completely divisible by a, b and c. There are different methods to calculate LCM of which three are the most important methods. They are:

1. LCM by Prime Factorisation.
2. LCM by Division.
3. LCM by listing the multiples. 

Let us learn LCM by prime factorization here. 

LCM by Prime Factorisation:

To calculate the LCM by the prime factorization method, we have to break down the numbers as a product of prime factors. The Prime factors are then written in exponential form. If the numbers have a common prime factor, then the common prime factor with the highest power is taken. The non-common prime factors are taken as-is along with their degrees. The common prime factors with the highest powers are multiplied with the non-common prime factors and the resulting product is the least common multiple. 

To better understand this method, let us find the LCM of 52 and 40 using the prime factorization method:

Step-1:

List out the prime factors of both the numbers.

52=2 \times 2 \times 13=2^{2} \times 13

40=2 \times 2 \times 2 \times 5=2^{3} \times 5

2 is a common prime factor of both the given numbers. So, only the highest power of 2 is considered. i.e., 2^{3}is considered and 2^{2}is ignored. The non-common prime factors being 13^{1}and 5^{1}are taken as-is. 

Step-2:

Multiply the prime factors with the highest powers. So we multiply 2^{3} \times 13^{1} \times 5^{1}.

Hence, the LCM is 2^{3} \times 13 \times 5=520.

Solved Examples:

1. Find LCM by prime factorization of 16, 12 and 18.

Solution:

First, let us list out the prime factors of the three numbers.

16=2 \times 2 \times 2 \times 2=2^{4}

Next, we multiply the prime factors with the highest powers. i.e. 2^{4} \times 3^{2}.

LCM of 16, 12, and 18 is 144.

2. Find the LCM of 25, 30 and 120

Solution:

Listing out the prime factors of the three given numbers, we get:

25=5 \times 5=5^{2}

120=2 \times 2 \times 2 \times 3 \times 5=2^{3} \times 3 \times 5

Multiplying the prime factors with the highest powers. i.e., 2^{3} \times 3 \times 5^{2}.

LCM of 25, 30, and 120 is 600.

3. Find the LCM of 11, 13, and 17.

Solution:

From the given question, we can infer that 11, 13 and 17 are prime numbers. The LCM of two or more prime numbers is simply the product of the prime numbers. 

Hence, the LCM of 11, 13 and 17 is 11 x 13 x 17 = 2431.