LCM of 30 and 90
LCM of 30 and 90 is the smallest number among all common multiples of 30 and 90. The first few multiples of 30 and 90 are (30, 60, 90, 120, 150, 180, 210, . . . ) and (90, 180, 270, 360, 450, 540, 630, . . . ) respectively. There are 3 commonly used methods to find LCM of 30 and 90  by division method, by listing multiples, and by prime factorization.
1.  LCM of 30 and 90 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 30 and 90?
Answer: LCM of 30 and 90 is 90.
Explanation:
The LCM of two nonzero integers, x(30) and y(90), is the smallest positive integer m(90) that is divisible by both x(30) and y(90) without any remainder.
Methods to Find LCM of 30 and 90
Let's look at the different methods for finding the LCM of 30 and 90.
 By Division Method
 By Listing Multiples
 By Prime Factorization Method
LCM of 30 and 90 by Division Method
To calculate the LCM of 30 and 90 by the division method, we will divide the numbers(30, 90) by their prime factors (preferably common). The product of these divisors gives the LCM of 30 and 90.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 30 and 90. Write this prime number(2) on the left of the given numbers(30 and 90), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (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 30 and 90 is the product of all prime numbers on the left, i.e. LCM(30, 90) by division method = 2 × 3 × 3 × 5 = 90.
LCM of 30 and 90 by Listing Multiples
To calculate the LCM of 30 and 90 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 30 (30, 60, 90, 120, 150, 180, 210, . . . ) and 90 (90, 180, 270, 360, 450, 540, 630, . . . . )
 Step 2: The common multiples from the multiples of 30 and 90 are 90, 180, . . .
 Step 3: The smallest common multiple of 30 and 90 is 90.
∴ The least common multiple of 30 and 90 = 90.
LCM of 30 and 90 by Prime Factorization
Prime factorization of 30 and 90 is (2 × 3 × 5) = 2^{1} × 3^{1} × 5^{1} and (2 × 3 × 3 × 5) = 2^{1} × 3^{2} × 5^{1} respectively. LCM of 30 and 90 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{1} × 3^{2} × 5^{1} = 90.
Hence, the LCM of 30 and 90 by prime factorization is 90.
☛ Also Check:
 LCM of 8 and 13  104
 LCM of 3, 6 and 7  42
 LCM of 4 and 22  44
 LCM of 3 and 6  6
 LCM of 45 and 75  225
 LCM of 13 and 52  52
 LCM of 16 and 28  112
LCM of 30 and 90 Examples

Example 1: The GCD and LCM of two numbers are 30 and 90 respectively. If one number is 30, find the other number.
Solution:
Let the other number be a.
∵ GCD × LCM = 30 × a
⇒ a = (GCD × LCM)/30
⇒ a = (30 × 90)/30
⇒ a = 90
Therefore, the other number is 90. 
Example 2: Verify the relationship between GCF and LCM of 30 and 90.
Solution:
The relation between GCF and LCM of 30 and 90 is given as,
LCM(30, 90) × GCF(30, 90) = Product of 30, 90
Prime factorization of 30 and 90 is given as, 30 = (2 × 3 × 5) = 2^{1} × 3^{1} × 5^{1} and 90 = (2 × 3 × 3 × 5) = 2^{1} × 3^{2} × 5^{1}
LCM(30, 90) = 90
GCF(30, 90) = 30
LHS = LCM(30, 90) × GCF(30, 90) = 90 × 30 = 2700
RHS = Product of 30, 90 = 30 × 90 = 2700
⇒ LHS = RHS = 2700
Hence, verified. 
Example 3: Find the smallest number that is divisible by 30 and 90 exactly.
Solution:
The smallest number that is divisible by 30 and 90 exactly is their LCM.
⇒ Multiples of 30 and 90: Multiples of 30 = 30, 60, 90, 120, 150, 180, 210, . . . .
 Multiples of 90 = 90, 180, 270, 360, 450, 540, 630, . . . .
Therefore, the LCM of 30 and 90 is 90.
FAQs on LCM of 30 and 90
What is the LCM of 30 and 90?
The LCM of 30 and 90 is 90. To find the least common multiple (LCM) of 30 and 90, we need to find the multiples of 30 and 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 30 and 90, i.e., 90.
What is the Relation Between GCF and LCM of 30, 90?
The following equation can be used to express the relation between GCF and LCM of 30 and 90, i.e. GCF × LCM = 30 × 90.
What are the Methods to Find LCM of 30 and 90?
The commonly used methods to find the LCM of 30 and 90 are:
 Division Method
 Prime Factorization Method
 Listing Multiples
If the LCM of 90 and 30 is 90, Find its GCF.
LCM(90, 30) × GCF(90, 30) = 90 × 30
Since the LCM of 90 and 30 = 90
⇒ 90 × GCF(90, 30) = 2700
Therefore, the GCF = 2700/90 = 30.
How to Find the LCM of 30 and 90 by Prime Factorization?
To find the LCM of 30 and 90 using prime factorization, we will find the prime factors, (30 = 2 × 3 × 5) and (90 = 2 × 3 × 3 × 5). LCM of 30 and 90 is the product of prime factors raised to their respective highest exponent among the numbers 30 and 90.
⇒ LCM of 30, 90 = 2^{1} × 3^{2} × 5^{1} = 90.