Question

Find the LCM of the following using prime factorization method.
$72,90,120$.

Answer 1

Hint: Here, we are required to find the L.C.M. of the given three numbers. We will find the factors of the three numbers by using the prime factorization method. Then multiplying all the factors of the three numbers by taking the highest power among the common factors, we will be able to the required L.C.M. of the three numbers.

Complete step-by-step answer:
First of all, by using the prime factorization method, we will find the factors of the given three numbers.
Hence, prime factorization of the first number 72 is:
$72 = 2 \times 2 \times 2 \times 3 \times 3 = {2^3} \times {3^2}$
Now, prime factorization of the second number 90 is:
$90 = 2 \times 3 \times 3 \times 5 = 2 \times {3^2} \times 5$
And, prime factorization of the third number 120 is:
$120 = 2 \times 2 \times 2 \times 3 \times 5 = {2^3} \times 3 \times 5$
Now, in order to find the L.C.M. we take all the factors present in three numbers and the highest power of the common factors respectively.
Hence, L.C.M. of these three numbers $ = {2^3} \times {3^2} \times 5 = 8 \times 9 \times 5 = 360$

Therefore, the required L.C.M. of $72,90,120$ is 360
Hence, this is the required answer.

Note:
In this question, we are required to express the given numbers as a product of their prime factors in order to find their LCM. Hence, we should know that prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Now, in order to express the given numbers as a product of their prime factors, we are required to do the prime factorization of the given numbers. Now, factorization is a method of writing an original number as the product of its various factors. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.