# Determine the LCM of $45,90$ and $30$.

A) 90

B) 180

C) 45

D) 20

Last updated date: 23rd Mar 2023

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Answer

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**Hint:**LCM (Least common Multiple) is the least or the smallest number with which the given numbers are exactly divisible. It is also known as the least common divisor. LCM can be expressed as the product of constant and HCF. Here first of all we will find the prime factors of the given two numbers and then LCM.

**Complete step by step solution:**

Find the prime factors of the given two or more numbers.

Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor. Here we will find the product of prime factors one by one for both the given numbers.

\[45 = 3 \times 3 \times 5\]

\[90 = 2 \times 3 \times 3 \times 5\]

\[30 = 2 \times 3 \times 5\]

LCM can be expressed as the product of the highest power of each factor involved in the numbers.

Therefore, the LCM of the given three numbers 45, 90 and 30 is $ = {2^1} \times {3^2} \times {5^1} = 90$

This is the required solution.

**Therefore, the LCM of $45,90$ and $30$ is $90$. So, the correct option is (A).**

**Note:**

To solve these types of sums, one should be clear about the concept of HCF and LCM and the prime numbers. HCF is the highest or greatest common multiple whereas the LCM is the least common multiple or least common divisor in two or more given numbers. Prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor. Factors are the number $1$ and the number itself. Also, remember that we get the prime factorization of any composite number.

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