LCM of 54 and 72
LCM of 54 and 72 is the smallest number among all common multiples of 54 and 72. The first few multiples of 54 and 72 are (54, 108, 162, 216, 270, 324, . . . ) and (72, 144, 216, 288, 360, . . . ) respectively. There are 3 commonly used methods to find LCM of 54 and 72  by listing multiples, by division method, and by prime factorization.
1.  LCM of 54 and 72 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 54 and 72?
Answer: LCM of 54 and 72 is 216.
Explanation:
The LCM of two nonzero integers, x(54) and y(72), is the smallest positive integer m(216) that is divisible by both x(54) and y(72) without any remainder.
Methods to Find LCM of 54 and 72
Let's look at the different methods for finding the LCM of 54 and 72.
 By Prime Factorization Method
 By Listing Multiples
 By Division Method
LCM of 54 and 72 by Prime Factorization
Prime factorization of 54 and 72 is (2 × 3 × 3 × 3) = 2^{1} × 3^{3} and (2 × 2 × 2 × 3 × 3) = 2^{3} × 3^{2} respectively. LCM of 54 and 72 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{3} × 3^{3} = 216.
Hence, the LCM of 54 and 72 by prime factorization is 216.
LCM of 54 and 72 by Listing Multiples
To calculate the LCM of 54 and 72 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 54 (54, 108, 162, 216, 270, 324, . . . ) and 72 (72, 144, 216, 288, 360, . . . . )
 Step 2: The common multiples from the multiples of 54 and 72 are 216, 432, . . .
 Step 3: The smallest common multiple of 54 and 72 is 216.
∴ The least common multiple of 54 and 72 = 216.
LCM of 54 and 72 by Division Method
To calculate the LCM of 54 and 72 by the division method, we will divide the numbers(54, 72) by their prime factors (preferably common). The product of these divisors gives the LCM of 54 and 72.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 54 and 72. Write this prime number(2) on the left of the given numbers(54 and 72), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (54, 72) 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 54 and 72 is the product of all prime numbers on the left, i.e. LCM(54, 72) by division method = 2 × 2 × 2 × 3 × 3 × 3 = 216.
☛ Also Check:
 LCM of 12 and 45  180
 LCM of 48, 56 and 72  1008
 LCM of 18, 24 and 30  360
 LCM of 6 and 9  18
 LCM of 72 and 24  72
 LCM of 12 and 40  120
 LCM of 24 and 40  120
LCM of 54 and 72 Examples

Example 1: The GCD and LCM of two numbers are 18 and 216 respectively. If one number is 72, find the other number.
Solution:
Let the other number be z.
∵ GCD × LCM = 72 × z
⇒ z = (GCD × LCM)/72
⇒ z = (18 × 216)/72
⇒ z = 54
Therefore, the other number is 54. 
Example 2: Verify the relationship between GCF and LCM of 54 and 72.
Solution:
The relation between GCF and LCM of 54 and 72 is given as,
LCM(54, 72) × GCF(54, 72) = Product of 54, 72
Prime factorization of 54 and 72 is given as, 54 = (2 × 3 × 3 × 3) = 2^{1} × 3^{3} and 72 = (2 × 2 × 2 × 3 × 3) = 2^{3} × 3^{2}
LCM(54, 72) = 216
GCF(54, 72) = 18
LHS = LCM(54, 72) × GCF(54, 72) = 216 × 18 = 3888
RHS = Product of 54, 72 = 54 × 72 = 3888
⇒ LHS = RHS = 3888
Hence, verified. 
Example 3: Find the smallest number that is divisible by 54 and 72 exactly.
Solution:
The smallest number that is divisible by 54 and 72 exactly is their LCM.
⇒ Multiples of 54 and 72: Multiples of 54 = 54, 108, 162, 216, 270, 324, 378, . . . .
 Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, . . . .
Therefore, the LCM of 54 and 72 is 216.
FAQs on LCM of 54 and 72
What is the LCM of 54 and 72?
The LCM of 54 and 72 is 216. To find the LCM (least common multiple) of 54 and 72, we need to find the multiples of 54 and 72 (multiples of 54 = 54, 108, 162, 216; multiples of 72 = 72, 144, 216, 288) and choose the smallest multiple that is exactly divisible by 54 and 72, i.e., 216.
What are the Methods to Find LCM of 54 and 72?
The commonly used methods to find the LCM of 54 and 72 are:
 Division Method
 Prime Factorization Method
 Listing Multiples
If the LCM of 72 and 54 is 216, Find its GCF.
LCM(72, 54) × GCF(72, 54) = 72 × 54
Since the LCM of 72 and 54 = 216
⇒ 216 × GCF(72, 54) = 3888
Therefore, the greatest common factor (GCF) = 3888/216 = 18.
What is the Least Perfect Square Divisible by 54 and 72?
The least number divisible by 54 and 72 = LCM(54, 72)
LCM of 54 and 72 = 2 × 2 × 2 × 3 × 3 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 54 and 72 = LCM(54, 72) × 2 × 3 = 1296 [Square root of 1296 = √1296 = ±36]
Therefore, 1296 is the required number.
How to Find the LCM of 54 and 72 by Prime Factorization?
To find the LCM of 54 and 72 using prime factorization, we will find the prime factors, (54 = 2 × 3 × 3 × 3) and (72 = 2 × 2 × 2 × 3 × 3). LCM of 54 and 72 is the product of prime factors raised to their respective highest exponent among the numbers 54 and 72.
⇒ LCM of 54, 72 = 2^{3} × 3^{3} = 216.
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