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LCM of 15, 30, and 90

LCM of 15, 30, and 90 is the smallest number among all common multiples of 15, 30, and 90. The first few multiples of 15, 30, and 90 are (15, 30, 45, 60, 75 . . .), (30, 60, 90, 120, 150 . . .), and (90, 180, 270, 360, 450 . . .) respectively. There are 3 commonly used methods to find LCM of 15, 30, 90 - by division method, by prime factorization, and by listing multiples.

1.	LCM of 15, 30, and 90
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 15, 30, and 90?

Answer: LCM of 15, 30, and 90 is 90.

Explanation:

The LCM of three non-zero integers, a(15), b(30), and c(90), is the smallest positive integer m(90) that is divisible by a(15), b(30), and c(90) without any remainder.

Methods to Find LCM of 15, 30, and 90

Let's look at the different methods for finding the LCM of 15, 30, and 90.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 15, 30, and 90 by Prime Factorization

Prime factorization of 15, 30, and 90 is (3 × 5) = 3¹ × 5¹, (2 × 3 × 5) = 2¹ × 3¹ × 5¹, and (2 × 3 × 3 × 5) = 2¹ × 3² × 5¹ respectively. LCM of 15, 30, and 90 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3² × 5¹ = 90.
Hence, the LCM of 15, 30, and 90 by prime factorization is 90.

LCM of 15, 30, and 90 by Listing Multiples

To calculate the LCM of 15, 30, 90 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 15 (15, 30, 45, 60, 75 . . .), 30 (30, 60, 90, 120, 150 . . .), and 90 (90, 180, 270, 360, 450 . . .).
Step 2: The common multiples from the multiples of 15, 30, and 90 are 90, 180, . . .
Step 3: The smallest common multiple of 15, 30, and 90 is 90.

∴ The least common multiple of 15, 30, and 90 = 90.

LCM of 15, 30, and 90 by Division Method

To calculate the LCM of 15, 30, and 90 by the division method, we will divide the numbers(15, 30, 90) by their prime factors (preferably common). The product of these divisors gives the LCM of 15, 30, and 90.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 15, 30, and 90. Write this prime number(2) on the left of the given numbers(15, 30, and 90), separated as per the ladder arrangement.
Step 2: If any of the given numbers (15, 30, 90) 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 15, 30, and 90 is the product of all prime numbers on the left, i.e. LCM(15, 30, 90) by division method = 2 × 3 × 3 × 5 = 90.

☛ Also Check:

LCM of 15, 30, and 90 Examples

Example 1: Find the smallest number that is divisible by 15, 30, 90 exactly.

Solution:

The smallest number that is divisible by 15, 30, and 90 exactly is their LCM.
⇒ Multiples of 15, 30, and 90:
- Multiples of 15 = 15, 30, 45, 60, 75, 90, . . . .
- Multiples of 30 = 30, 60, 90, 120, 150, 180, . . . .
- Multiples of 90 = 90, 180, 270, 360, 450, 540, . . . .
Therefore, the LCM of 15, 30, and 90 is 90.
Example 2: Calculate the LCM of 15, 30, and 90 using the GCD of the given numbers.

Solution:

Prime factorization of 15, 30, 90:
- 15 = 3¹ × 5¹
- 30 = 2¹ × 3¹ × 5¹
- 90 = 2¹ × 3² × 5¹
Therefore, GCD(15, 30) = 15, GCD(30, 90) = 30, GCD(15, 90) = 15, GCD(15, 30, 90) = 15
We know,
LCM(15, 30, 90) = [(15 × 30 × 90) × GCD(15, 30, 90)]/[GCD(15, 30) × GCD(30, 90) × GCD(15, 90)]
LCM(15, 30, 90) = (40500 × 15)/(15 × 30 × 15) = 90
⇒LCM(15, 30, 90) = 90
Example 3: Verify the relationship between the GCD and LCM of 15, 30, and 90.

Solution:

The relation between GCD and LCM of 15, 30, and 90 is given as,
LCM(15, 30, 90) = [(15 × 30 × 90) × GCD(15, 30, 90)]/[GCD(15, 30) × GCD(30, 90) × GCD(15, 90)]
⇒ Prime factorization of 15, 30 and 90:
- 15 = 3¹ × 5¹
- 30 = 2¹ × 3¹ × 5¹
- 90 = 2¹ × 3² × 5¹
∴ GCD of (15, 30), (30, 90), (15, 90) and (15, 30, 90) = 15, 30, 15 and 15 respectively.
Now, LHS = LCM(15, 30, 90) = 90.
And, RHS = [(15 × 30 × 90) × GCD(15, 30, 90)]/[GCD(15, 30) × GCD(30, 90) × GCD(15, 90)] = [(40500) × 15]/[15 × 30 × 15] = 90
LHS = RHS = 90.
Hence verified.

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FAQs on LCM of 15, 30, and 90

What is the LCM of 15, 30, and 90?

The LCM of 15, 30, and 90 is 90. To find the least common multiple of 15, 30, and 90, we need to find the multiples of 15, 30, and 90 (multiples of 15 = 15, 30, 45, 60, 90 . . . .; multiples of 30 = 30, 60, 90, 120 . . . .; multiples of 90 = 90, 180, 270, 360 . . . .) and choose the smallest multiple that is exactly divisible by 15, 30, and 90, i.e., 90.

What is the Relation Between GCF and LCM of 15, 30, 90?

The following equation can be used to express the relation between GCF and LCM of 15, 30, 90, i.e. LCM(15, 30, 90) = [(15 × 30 × 90) × GCF(15, 30, 90)]/[GCF(15, 30) × GCF(30, 90) × GCF(15, 90)].

What are the Methods to Find LCM of 15, 30, 90?

The commonly used methods to find the LCM of 15, 30, 90 are:

Prime Factorization Method
Division Method
Listing Multiples

What is the Least Perfect Square Divisible by 15, 30, and 90?

The least number divisible by 15, 30, and 90 = LCM(15, 30, 90)
LCM of 15, 30, and 90 = 2 × 3 × 3 × 5 [Incomplete pair(s): 2, 5]
⇒ Least perfect square divisible by each 15, 30, and 90 = LCM(15, 30, 90) × 2 × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

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