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LCM of 60, 84, and 108

LCM of 60, 84, and 108 is the smallest number among all common multiples of 60, 84, and 108. The first few multiples of 60, 84, and 108 are (60, 120, 180, 240, 300 . . .), (84, 168, 252, 336, 420 . . .), and (108, 216, 324, 432, 540 . . .) respectively. There are 3 commonly used methods to find LCM of 60, 84, 108 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 60, 84, and 108
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 60, 84, and 108?

Answer: LCM of 60, 84, and 108 is 3780.

Explanation:

The LCM of three non-zero integers, a(60), b(84), and c(108), is the smallest positive integer m(3780) that is divisible by a(60), b(84), and c(108) without any remainder.

Methods to Find LCM of 60, 84, and 108

The methods to find the LCM of 60, 84, and 108 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 60, 84, and 108 by Listing Multiples

To calculate the LCM of 60, 84, 108 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 60 (60, 120, 180, 240, 300 . . .), 84 (84, 168, 252, 336, 420 . . .), and 108 (108, 216, 324, 432, 540 . . .).
Step 2: The common multiples from the multiples of 60, 84, and 108 are 3780, 7560, . . .
Step 3: The smallest common multiple of 60, 84, and 108 is 3780.

∴ The least common multiple of 60, 84, and 108 = 3780.

LCM of 60, 84, and 108 by Prime Factorization

Prime factorization of 60, 84, and 108 is (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹, (2 × 2 × 3 × 7) = 2² × 3¹ × 7¹, and (2 × 2 × 3 × 3 × 3) = 2² × 3³ respectively. LCM of 60, 84, and 108 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3³ × 5¹ × 7¹ = 3780.
Hence, the LCM of 60, 84, and 108 by prime factorization is 3780.

LCM of 60, 84, and 108 by Division Method

To calculate the LCM of 60, 84, and 108 by the division method, we will divide the numbers(60, 84, 108) by their prime factors (preferably common). The product of these divisors gives the LCM of 60, 84, and 108.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 60, 84, and 108. Write this prime number(2) on the left of the given numbers(60, 84, and 108), separated as per the ladder arrangement.
Step 2: If any of the given numbers (60, 84, 108) 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 60, 84, and 108 is the product of all prime numbers on the left, i.e. LCM(60, 84, 108) by division method = 2 × 2 × 3 × 3 × 3 × 5 × 7 = 3780.

☛ Also Check:

LCM of 60, 84, and 108 Examples

Example 1: Calculate the LCM of 60, 84, and 108 using the GCD of the given numbers.

Solution:

Prime factorization of 60, 84, 108:
- 60 = 2² × 3¹ × 5¹
- 84 = 2² × 3¹ × 7¹
- 108 = 2² × 3³
Therefore, GCD(60, 84) = 12, GCD(84, 108) = 12, GCD(60, 108) = 12, GCD(60, 84, 108) = 12
We know,
LCM(60, 84, 108) = [(60 × 84 × 108) × GCD(60, 84, 108)]/[GCD(60, 84) × GCD(84, 108) × GCD(60, 108)]
LCM(60, 84, 108) = (544320 × 12)/(12 × 12 × 12) = 3780
⇒LCM(60, 84, 108) = 3780
Example 2: Find the smallest number that is divisible by 60, 84, 108 exactly.

Solution:

The value of LCM(60, 84, 108) will be the smallest number that is exactly divisible by 60, 84, and 108.
⇒ Multiples of 60, 84, and 108:
- Multiples of 60 = 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, . . . ., 3600, 3660, 3720, 3780, . . . .
- Multiples of 84 = 84, 168, 252, 336, 420, 504, 588, 672, 756, 840, . . . ., 3444, 3528, 3612, 3696, 3780, . . . .
- Multiples of 108 = 108, 216, 324, 432, 540, 648, 756, 864, 972, 1080, . . . ., 3564, 3672, 3780, . . . .
Therefore, the LCM of 60, 84, and 108 is 3780.
Example 3: Verify the relationship between the GCD and LCM of 60, 84, and 108.

Solution:

The relation between GCD and LCM of 60, 84, and 108 is given as,
LCM(60, 84, 108) = [(60 × 84 × 108) × GCD(60, 84, 108)]/[GCD(60, 84) × GCD(84, 108) × GCD(60, 108)]
⇒ Prime factorization of 60, 84 and 108:
- 60 = 2² × 3¹ × 5¹
- 84 = 2² × 3¹ × 7¹
- 108 = 2² × 3³
∴ GCD of (60, 84), (84, 108), (60, 108) and (60, 84, 108) = 12, 12, 12 and 12 respectively.
Now, LHS = LCM(60, 84, 108) = 3780.
And, RHS = [(60 × 84 × 108) × GCD(60, 84, 108)]/[GCD(60, 84) × GCD(84, 108) × GCD(60, 108)] = [(544320) × 12]/[12 × 12 × 12] = 3780
LHS = RHS = 3780.
Hence verified.

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FAQs on LCM of 60, 84, and 108

What is the LCM of 60, 84, and 108?

The LCM of 60, 84, and 108 is 3780. To find the LCM of 60, 84, and 108, we need to find the multiples of 60, 84, and 108 (multiples of 60 = 60, 120, 180, 240 . . . . 3780 . . . . ; multiples of 84 = 84, 168, 252, 336 . . . . 3780 . . . . ; multiples of 108 = 108, 216, 324, 432 . . . . 3780 . . . . ) and choose the smallest multiple that is exactly divisible by 60, 84, and 108, i.e., 3780.

Which of the following is the LCM of 60, 84, and 108? 35, 45, 32, 3780

The value of LCM of 60, 84, 108 is the smallest common multiple of 60, 84, and 108. The number satisfying the given condition is 3780.

How to Find the LCM of 60, 84, and 108 by Prime Factorization?

To find the LCM of 60, 84, and 108 using prime factorization, we will find the prime factors, (60 = 2² × 3¹ × 5¹), (84 = 2² × 3¹ × 7¹), and (108 = 2² × 3³). LCM of 60, 84, and 108 is the product of prime factors raised to their respective highest exponent among the numbers 60, 84, and 108.
⇒ LCM of 60, 84, 108 = 2² × 3³ × 5¹ × 7¹ = 3780.

What is the Relation Between GCF and LCM of 60, 84, 108?

The following equation can be used to express the relation between GCF and LCM of 60, 84, 108, i.e. LCM(60, 84, 108) = [(60 × 84 × 108) × GCF(60, 84, 108)]/[GCF(60, 84) × GCF(84, 108) × GCF(60, 108)].

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