Question

Find the LCM of 84, 90 and 120 by applying the prime factorization method.

Answer 1

Hint: Here we need to determine the LCM of the given three numbers. First, we will express each number in terms of the product of its prime factors. Then we will multiply all prime factors that have occurred the maximum number of times in either of the numbers. This is called the prime factorization method.

Complete step-by-step answer:
Here we need to find the LCM of the numbers 84, 90 and 120 using the method of prime factorization.
Now, we will first express each number in terms of the product of its prime factors.
To write the prime factors, we should always start with the smallest prime number i.e. 2 and check divisibility. If the number is divisible by the prime number, then we can write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by that prime number. Then we take the quotient and we will repeat the same process. This process is repeated until we get the quotient as 1.
Now, we will find the factors of the first number 84.
We know that 84 is an even number, so it can be written as
\[84 = 2 \times 42\]
Now, we will further break the number 42 into its factors.
\[ \Rightarrow 84 = 2 \times 2 \times 3 \times 7\]
Similarly, we will find the factors of the second number 90.
We know that 90 is an even number, so it can be written as
\[90 = 2 \times 45\]
Now, we will further break the number 45 into its factors.
\[ \Rightarrow 90 = 2 \times 3 \times 3 \times 5\]
Again we will find the factors of the third number 120.
We know that 120 is an even number, so it can be written as
\[120 = 2 \times 60\]
Now, we will further break the number 60 into its factors as we can see that it is a multiple of 6 and 10.
Therefore, we can write it as
\[ \Rightarrow 120 = 2 \times 2 \times 2 \times 3 \times 5\]
Now, to find the LCM, we need to multiply all the prime factors that have occurred maximum number of times in either of the numbers. So, LCM of 84, 90, and 120 is a product of three 2, one 5 and one 7. As we can see that 2 have occurred three times in 120, 3 is occurring a maximum number of 2 times and 5 and 7 are occurring a maximum number of one time.
Therefore,
\[LCM\left( {80,90,120} \right) = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7\]
On multiplying the number, we get
\[ \Rightarrow LCM\left( {80,90,120} \right) = 2520\]


Note: Here, we might get confused between factors of a number and multiples of a number. A factor is defined as a number which divides the given number completely but the multiple is defined as a number that is completely divided by the given number. Let’s take the number 10 as an example. We know that the factors of number 10 are 1, 2, 5, and 10 but the multiples of number 10 are 10, 20, 30, 40, and so on.