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LCM of 36 and 84

LCM of 36 and 84 is the smallest number among all common multiples of 36 and 84. The first few multiples of 36 and 84 are (36, 72, 108, 144, 180, 216, 252, . . . ) and (84, 168, 252, 336, 420, 504, 588, . . . ) respectively. There are 3 commonly used methods to find LCM of 36 and 84 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 36 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 36 and 84?

Answer: LCM of 36 and 84 is 252.

Explanation:

The LCM of two non-zero integers, x(36) and y(84), is the smallest positive integer m(252) that is divisible by both x(36) and y(84) without any remainder.

Methods to Find LCM of 36 and 84

The methods to find the LCM of 36 and 84 are explained below.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 36 and 84 by Listing Multiples

To calculate the LCM of 36 and 84 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 36 (36, 72, 108, 144, 180, 216, 252, . . . ) and 84 (84, 168, 252, 336, 420, 504, 588, . . . . )
Step 2: The common multiples from the multiples of 36 and 84 are 252, 504, . . .
Step 3: The smallest common multiple of 36 and 84 is 252.

∴ The least common multiple of 36 and 84 = 252.

LCM of 36 and 84 by Division Method

To calculate the LCM of 36 and 84 by the division method, we will divide the numbers(36, 84) by their prime factors (preferably common). The product of these divisors gives the LCM of 36 and 84.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 36 and 84. Write this prime number(2) on the left of the given numbers(36 and 84), separated as per the ladder arrangement.
Step 2: If any of the given numbers (36, 84) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 36 and 84 is the product of all prime numbers on the left, i.e. LCM(36, 84) by division method = 2 × 2 × 3 × 3 × 7 = 252.

LCM of 36 and 84 by Prime Factorization

Prime factorization of 36 and 84 is (2 × 2 × 3 × 3) = 2² × 3² and (2 × 2 × 3 × 7) = 2² × 3¹ × 7¹ respectively. LCM of 36 and 84 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 7¹ = 252.
Hence, the LCM of 36 and 84 by prime factorization is 252.

☛ Also Check:

LCM of 36 and 84 Examples

Example 1: The GCD and LCM of two numbers are 12 and 252 respectively. If one number is 84, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 84 × y
⇒ y = (GCD × LCM)/84
⇒ y = (12 × 252)/84
⇒ y = 36
Therefore, the other number is 36.
Example 2: Verify the relationship between GCF and LCM of 36 and 84.

Solution:

The relation between GCF and LCM of 36 and 84 is given as,
LCM(36, 84) × GCF(36, 84) = Product of 36, 84
Prime factorization of 36 and 84 is given as, 36 = (2 × 2 × 3 × 3) = 2² × 3² and 84 = (2 × 2 × 3 × 7) = 2² × 3¹ × 7¹
LCM(36, 84) = 252
GCF(36, 84) = 12
LHS = LCM(36, 84) × GCF(36, 84) = 252 × 12 = 3024
RHS = Product of 36, 84 = 36 × 84 = 3024
⇒ LHS = RHS = 3024
Hence, verified.
Example 3: The product of two numbers is 3024. If their GCD is 12, what is their LCM?

Solution:

Given: GCD = 12
product of numbers = 3024
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 3024/12
Therefore, the LCM is 252.
The probable combination for the given case is LCM(36, 84) = 252.

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FAQs on LCM of 36 and 84

What is the LCM of 36 and 84?

The LCM of 36 and 84 is 252. To find the LCM (least common multiple) of 36 and 84, we need to find the multiples of 36 and 84 (multiples of 36 = 36, 72, 108, 144 . . . . 252; multiples of 84 = 84, 168, 252, 336) and choose the smallest multiple that is exactly divisible by 36 and 84, i.e., 252.

What is the Least Perfect Square Divisible by 36 and 84?

The least number divisible by 36 and 84 = LCM(36, 84)
LCM of 36 and 84 = 2 × 2 × 3 × 3 × 7 [Incomplete pair(s): 7]
⇒ Least perfect square divisible by each 36 and 84 = LCM(36, 84) × 7 = 1764 [Square root of 1764 = √1764 = ±42]
Therefore, 1764 is the required number.

What is the Relation Between GCF and LCM of 36, 84?

The following equation can be used to express the relation between GCF and LCM of 36 and 84, i.e. GCF × LCM = 36 × 84.

If the LCM of 84 and 36 is 252, Find its GCF.

LCM(84, 36) × GCF(84, 36) = 84 × 36
Since the LCM of 84 and 36 = 252
⇒ 252 × GCF(84, 36) = 3024
Therefore, the greatest common factor = 3024/252 = 12.

How to Find the LCM of 36 and 84 by Prime Factorization?

To find the LCM of 36 and 84 using prime factorization, we will find the prime factors, (36 = 2 × 2 × 3 × 3) and (84 = 2 × 2 × 3 × 7). LCM of 36 and 84 is the product of prime factors raised to their respective highest exponent among the numbers 36 and 84.
⇒ LCM of 36, 84 = 2² × 3² × 7¹ = 252.

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