Question

Find the LCM of 84, 90 and 120.

Answer 1

- Hint:To determine the LCM of the numbers, express the number in terms of the product of its prime factors and multiply all the prime factors the maximum number of times they occur in either number. This is the method of prime factorization.

Complete step-by-step solution -

Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, 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 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 the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an odd number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now starting with finding the factors of 84. We know 84 is an even number, so it can be written as $2=2\times 42$ . Further we can break 42 as $42=2\times 7\times 3$ . Therefore, we can write 84 as $2\times 2\times 3\times 7$ .
Now let us move to the factorisation of 90. So, as 90 is a multiple of 9 and 10 , it must have $3$ as one of its factors. Also, 90 is divisible by 2 and 5, i.e., 10 . So, 90 can be written as $90=2\times 3\times 3\times 5$ .
Now we will find the factors of 120. We know 120 is divisible by 12 and 10. Also, we know a number divisible by 10 must have 2 and 5 as its prime factors. Also, being divisible by 12, makes two 2s and 3 as its factors. So, 120 can be written as $120=2\times 2\times 2\times 3\times 5$ .
Now to find the LCM, we need to multiply all the prime factors the maximum number of times they occur in either number. So, LCM(84,90,120) is a product of three 2s, one 5, one 7 and two 3s, as 2 is occurring 3 times in 120 and 3 is occurring a maximum of 2 times in 90 and 5 and 7 are occurring a maximum of one time.
$LCM\left( 84,90,120 \right)=2\times 2\times 2\times 3\times 3\times 5\times 7=2520.$
Therefore, we can conclude that the LCM of 84, 90 and 120 is 2520.

Note: Be careful while finding the prime factors of each number. Also, it is prescribed that you learn the division method of finding the LCM as well, as it might be helpful. If in case you are asked to find the LCM of two fractions you must use the formula $LCM=\dfrac{LCM\text{ of numerator of the fractions}}{HCF\text{ of the denominator of the fractions}}$ .