LCM of 15, 20, and 30
LCM of 15, 20, and 30 is the smallest number among all common multiples of 15, 20, and 30. The first few multiples of 15, 20, and 30 are (15, 30, 45, 60, 75 . . .), (20, 40, 60, 80, 100 . . .), and (30, 60, 90, 120, 150 . . .) respectively. There are 3 commonly used methods to find LCM of 15, 20, 30  by prime factorization, by listing multiples, and by division method.
1.  LCM of 15, 20, and 30 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 15, 20, and 30?
Answer: LCM of 15, 20, and 30 is 60.
Explanation:
The LCM of three nonzero integers, a(15), b(20), and c(30), is the smallest positive integer m(60) that is divisible by a(15), b(20), and c(30) without any remainder.
Methods to Find LCM of 15, 20, and 30
The methods to find the LCM of 15, 20, and 30 are explained below.
 By Division Method
 By Listing Multiples
 By Prime Factorization Method
LCM of 15, 20, and 30 by Division Method
To calculate the LCM of 15, 20, and 30 by the division method, we will divide the numbers(15, 20, 30) by their prime factors (preferably common). The product of these divisors gives the LCM of 15, 20, and 30.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 15, 20, and 30. Write this prime number(2) on the left of the given numbers(15, 20, and 30), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (15, 20, 30) 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, 20, and 30 is the product of all prime numbers on the left, i.e. LCM(15, 20, 30) by division method = 2 × 2 × 3 × 5 = 60.
LCM of 15, 20, and 30 by Listing Multiples
To calculate the LCM of 15, 20, 30 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 . . .), 20 (20, 40, 60, 80, 100 . . .), and 30 (30, 60, 90, 120, 150 . . .).
 Step 2: The common multiples from the multiples of 15, 20, and 30 are 60, 120, . . .
 Step 3: The smallest common multiple of 15, 20, and 30 is 60.
∴ The least common multiple of 15, 20, and 30 = 60.
LCM of 15, 20, and 30 by Prime Factorization
Prime factorization of 15, 20, and 30 is (3 × 5) = 3^{1} × 5^{1}, (2 × 2 × 5) = 2^{2} × 5^{1}, and (2 × 3 × 5) = 2^{1} × 3^{1} × 5^{1} respectively. LCM of 15, 20, and 30 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{1} × 5^{1} = 60.
Hence, the LCM of 15, 20, and 30 by prime factorization is 60.
ā Also Check:
 LCM of 148 and 185  740
 LCM of 20, 40 and 60  120
 LCM of 25 and 16  400
 LCM of 16 and 24  48
 LCM of 200 and 300  600
 LCM of 72 and 108  216
 LCM of 3 and 7  21
LCM of 15, 20, and 30 Examples

Example 1: Calculate the LCM of 15, 20, and 30 using the GCD of the given numbers.
Solution:
Prime factorization of 15, 20, 30:
 15 = 3^{1} × 5^{1}
 20 = 2^{2} × 5^{1}
 30 = 2^{1} × 3^{1} × 5^{1}
Therefore, GCD(15, 20) = 5, GCD(20, 30) = 10, GCD(15, 30) = 15, GCD(15, 20, 30) = 5
We know,
LCM(15, 20, 30) = [(15 × 20 × 30) × GCD(15, 20, 30)]/[GCD(15, 20) × GCD(20, 30) × GCD(15, 30)]
LCM(15, 20, 30) = (9000 × 5)/(5 × 10 × 15) = 60
⇒LCM(15, 20, 30) = 60 
Example 2: Verify the relationship between the GCD and LCM of 15, 20, and 30.
Solution:
The relation between GCD and LCM of 15, 20, and 30 is given as,
LCM(15, 20, 30) = [(15 × 20 × 30) × GCD(15, 20, 30)]/[GCD(15, 20) × GCD(20, 30) × GCD(15, 30)]
⇒ Prime factorization of 15, 20 and 30: 15 = 3^{1} × 5^{1}
 20 = 2^{2} × 5^{1}
 30 = 2^{1} × 3^{1} × 5^{1}
∴ GCD of (15, 20), (20, 30), (15, 30) and (15, 20, 30) = 5, 10, 15 and 5 respectively.
Now, LHS = LCM(15, 20, 30) = 60.
And, RHS = [(15 × 20 × 30) × GCD(15, 20, 30)]/[GCD(15, 20) × GCD(20, 30) × GCD(15, 30)] = [(9000) × 5]/[5 × 10 × 15] = 60
LHS = RHS = 60.
Hence verified. 
Example 3: Find the smallest number that is divisible by 15, 20, 30 exactly.
Solution:
The smallest number that is divisible by 15, 20, and 30 exactly is their LCM.
⇒ Multiples of 15, 20, and 30: Multiples of 15 = 15, 30, 45, 60, 75, 90, . . . .
 Multiples of 20 = 20, 40, 60, 80, 100, 120, . . . .
 Multiples of 30 = 30, 60, 90, 120, 150, 180, . . . .
Therefore, the LCM of 15, 20, and 30 is 60.
FAQs on LCM of 15, 20, and 30
What is the LCM of 15, 20, and 30?
The LCM of 15, 20, and 30 is 60. To find the least common multiple of 15, 20, and 30, we need to find the multiples of 15, 20, and 30 (multiples of 15 = 15, 30, 45 . . . .; multiples of 20 = 20, 40, 60 . . . .; multiples of 30 = 30, 60, 90 . . . .) and choose the smallest multiple that is exactly divisible by 15, 20, and 30, i.e., 60.
What are the Methods to Find LCM of 15, 20, 30?
The commonly used methods to find the LCM of 15, 20, 30 are:
 Prime Factorization Method
 Division Method
 Listing Multiples
What is the Least Perfect Square Divisible by 15, 20, and 30?
The least number divisible by 15, 20, and 30 = LCM(15, 20, 30)
LCM of 15, 20, and 30 = 2 × 2 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 15, 20, and 30 = LCM(15, 20, 30) × 3 × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.
Which of the following is the LCM of 15, 20, and 30? 15, 60, 36, 3
The value of LCM of 15, 20, 30 is the smallest common multiple of 15, 20, and 30. The number satisfying the given condition is 60.
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