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LCM of 84, 90, and 120

LCM of 84, 90, and 120 is the smallest number among all common multiples of 84, 90, and 120. The first few multiples of 84, 90, and 120 are (84, 168, 252, 336, 420 . . .), (90, 180, 270, 360, 450 . . .), and (120, 240, 360, 480, 600 . . .) respectively. There are 3 commonly used methods to find LCM of 84, 90, 120 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 84, 90, and 120
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 84, 90, and 120?

Answer: LCM of 84, 90, and 120 is 2520.

Explanation:

The LCM of three non-zero integers, a(84), b(90), and c(120), is the smallest positive integer m(2520) that is divisible by a(84), b(90), and c(120) without any remainder.

Methods to Find LCM of 84, 90, and 120

Let's look at the different methods for finding the LCM of 84, 90, and 120.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 84, 90, and 120 by Division Method

To calculate the LCM of 84, 90, and 120 by the division method, we will divide the numbers(84, 90, 120) by their prime factors (preferably common). The product of these divisors gives the LCM of 84, 90, and 120.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 84, 90, and 120. Write this prime number(2) on the left of the given numbers(84, 90, and 120), separated as per the ladder arrangement.
Step 2: If any of the given numbers (84, 90, 120) 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 84, 90, and 120 is the product of all prime numbers on the left, i.e. LCM(84, 90, 120) by division method = 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2520.

LCM of 84, 90, and 120 by Listing Multiples

To calculate the LCM of 84, 90, 120 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 84 (84, 168, 252, 336, 420 . . .), 90 (90, 180, 270, 360, 450 . . .), and 120 (120, 240, 360, 480, 600 . . .).
Step 2: The common multiples from the multiples of 84, 90, and 120 are 2520, 5040, . . .
Step 3: The smallest common multiple of 84, 90, and 120 is 2520.

∴ The least common multiple of 84, 90, and 120 = 2520.

LCM of 84, 90, and 120 by Prime Factorization

Prime factorization of 84, 90, and 120 is (2 × 2 × 3 × 7) = 2² × 3¹ × 7¹, (2 × 3 × 3 × 5) = 2¹ × 3² × 5¹, and (2 × 2 × 2 × 3 × 5) = 2³ × 3¹ × 5¹ respectively. LCM of 84, 90, and 120 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ × 7¹ = 2520.
Hence, the LCM of 84, 90, and 120 by prime factorization is 2520.

☛ Also Check:

LCM of 84, 90, and 120 Examples

Example 1: Calculate the LCM of 84, 90, and 120 using the GCD of the given numbers.

Solution:

Prime factorization of 84, 90, 120:
- 84 = 2² × 3¹ × 7¹
- 90 = 2¹ × 3² × 5¹
- 120 = 2³ × 3¹ × 5¹
Therefore, GCD(84, 90) = 6, GCD(90, 120) = 30, GCD(84, 120) = 12, GCD(84, 90, 120) = 6
We know,
LCM(84, 90, 120) = [(84 × 90 × 120) × GCD(84, 90, 120)]/[GCD(84, 90) × GCD(90, 120) × GCD(84, 120)]
LCM(84, 90, 120) = (907200 × 6)/(6 × 30 × 12) = 2520
⇒LCM(84, 90, 120) = 2520
Example 2: Find the smallest number that is divisible by 84, 90, 120 exactly.

Solution:

The value of LCM(84, 90, 120) will be the smallest number that is exactly divisible by 84, 90, and 120.
⇒ Multiples of 84, 90, and 120:
- Multiples of 84 = 84, 168, 252, 336, 420, 504, 588, 672, 756, 840, . . . ., 2184, 2268, 2352, 2436, 2520, . . . .
- Multiples of 90 = 90, 180, 270, 360, 450, 540, 630, 720, 810, 900, . . . ., 2250, 2340, 2430, 2520, . . . .
- Multiples of 120 = 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, . . . ., 2040, 2160, 2280, 2400, 2520, . . . .
Therefore, the LCM of 84, 90, and 120 is 2520.
Example 3: Verify the relationship between the GCD and LCM of 84, 90, and 120.

Solution:

The relation between GCD and LCM of 84, 90, and 120 is given as,
LCM(84, 90, 120) = [(84 × 90 × 120) × GCD(84, 90, 120)]/[GCD(84, 90) × GCD(90, 120) × GCD(84, 120)]
⇒ Prime factorization of 84, 90 and 120:
- 84 = 2² × 3¹ × 7¹
- 90 = 2¹ × 3² × 5¹
- 120 = 2³ × 3¹ × 5¹
∴ GCD of (84, 90), (90, 120), (84, 120) and (84, 90, 120) = 6, 30, 12 and 6 respectively.
Now, LHS = LCM(84, 90, 120) = 2520.
And, RHS = [(84 × 90 × 120) × GCD(84, 90, 120)]/[GCD(84, 90) × GCD(90, 120) × GCD(84, 120)] = [(907200) × 6]/[6 × 30 × 12] = 2520
LHS = RHS = 2520.
Hence verified.

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FAQs on LCM of 84, 90, and 120

What is the LCM of 84, 90, and 120?

The LCM of 84, 90, and 120 is 2520. To find the LCM (least common multiple) of 84, 90, and 120, we need to find the multiples of 84, 90, and 120 (multiples of 84 = 84, 168, 252, 336 . . . . 2520 . . . . ; multiples of 90 = 90, 180, 270, 360 . . . . 2520 . . . . ; multiples of 120 = 120, 240, 360, 480 . . . . 2520 . . . . ) and choose the smallest multiple that is exactly divisible by 84, 90, and 120, i.e., 2520.

Which of the following is the LCM of 84, 90, and 120? 32, 10, 5, 2520

The value of LCM of 84, 90, 120 is the smallest common multiple of 84, 90, and 120. The number satisfying the given condition is 2520.

How to Find the LCM of 84, 90, and 120 by Prime Factorization?

To find the LCM of 84, 90, and 120 using prime factorization, we will find the prime factors, (84 = 2² × 3¹ × 7¹), (90 = 2¹ × 3² × 5¹), and (120 = 2³ × 3¹ × 5¹). LCM of 84, 90, and 120 is the product of prime factors raised to their respective highest exponent among the numbers 84, 90, and 120.
⇒ LCM of 84, 90, 120 = 2³ × 3² × 5¹ × 7¹ = 2520.

What is the Least Perfect Square Divisible by 84, 90, and 120?

The least number divisible by 84, 90, and 120 = LCM(84, 90, 120)
LCM of 84, 90, and 120 = 2 × 2 × 2 × 3 × 3 × 5 × 7 [Incomplete pair(s): 2, 5, 7]
⇒ Least perfect square divisible by each 84, 90, and 120 = LCM(84, 90, 120) × 2 × 5 × 7 = 176400 [Square root of 176400 = √176400 = ±420]
Therefore, 176400 is the required number.

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