LCM of 144, 180, and 192
LCM of 144, 180, and 192 is the smallest number among all common multiples of 144, 180, and 192. The first few multiples of 144, 180, and 192 are (144, 288, 432, 576, 720 . . .), (180, 360, 540, 720, 900 . . .), and (192, 384, 576, 768, 960 . . .) respectively. There are 3 commonly used methods to find LCM of 144, 180, 192  by division method, by prime factorization, and by listing multiples.
1.  LCM of 144, 180, and 192 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 144, 180, and 192?
Answer: LCM of 144, 180, and 192 is 2880.
Explanation:
The LCM of three nonzero integers, a(144), b(180), and c(192), is the smallest positive integer m(2880) that is divisible by a(144), b(180), and c(192) without any remainder.
Methods to Find LCM of 144, 180, and 192
The methods to find the LCM of 144, 180, and 192 are explained below.
 By Prime Factorization Method
 By Division Method
 By Listing Multiples
LCM of 144, 180, and 192 by Prime Factorization
Prime factorization of 144, 180, and 192 is (2 × 2 × 2 × 2 × 3 × 3) = 2^{4} × 3^{2}, (2 × 2 × 3 × 3 × 5) = 2^{2} × 3^{2} × 5^{1}, and (2 × 2 × 2 × 2 × 2 × 2 × 3) = 2^{6} × 3^{1} respectively. LCM of 144, 180, and 192 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{6} × 3^{2} × 5^{1} = 2880.
Hence, the LCM of 144, 180, and 192 by prime factorization is 2880.
LCM of 144, 180, and 192 by Division Method
To calculate the LCM of 144, 180, and 192 by the division method, we will divide the numbers(144, 180, 192) by their prime factors (preferably common). The product of these divisors gives the LCM of 144, 180, and 192.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 144, 180, and 192. Write this prime number(2) on the left of the given numbers(144, 180, and 192), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (144, 180, 192) 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 144, 180, and 192 is the product of all prime numbers on the left, i.e. LCM(144, 180, 192) by division method = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2880.
LCM of 144, 180, and 192 by Listing Multiples
To calculate the LCM of 144, 180, 192 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 144 (144, 288, 432, 576, 720 . . .), 180 (180, 360, 540, 720, 900 . . .), and 192 (192, 384, 576, 768, 960 . . .).
 Step 2: The common multiples from the multiples of 144, 180, and 192 are 2880, 5760, . . .
 Step 3: The smallest common multiple of 144, 180, and 192 is 2880.
∴ The least common multiple of 144, 180, and 192 = 2880.
☛ Also Check:
 LCM of 2, 5 and 7  70
 LCM of 14 and 20  140
 LCM of 8 and 32  32
 LCM of 12, 18 and 20  180
 LCM of 36, 54 and 72  216
 LCM of 27 and 81  81
 LCM of 3 and 13  39
LCM of 144, 180, and 192 Examples

Example 1: Find the smallest number that is divisible by 144, 180, 192 exactly.
Solution:
The value of LCM(144, 180, 192) will be the smallest number that is exactly divisible by 144, 180, and 192.
⇒ Multiples of 144, 180, and 192: Multiples of 144 = 144, 288, 432, 576, 720, 864, 1008, 1152, 1296, 1440, . . . ., 2448, 2592, 2736, 2880, . . . .
 Multiples of 180 = 180, 360, 540, 720, 900, 1080, 1260, 1440, 1620, 1800, . . . ., 2340, 2520, 2700, 2880, . . . .
 Multiples of 192 = 192, 384, 576, 768, 960, 1152, 1344, 1536, 1728, 1920, . . . ., 2304, 2496, 2688, 2880, . . . .
Therefore, the LCM of 144, 180, and 192 is 2880.

Example 2: Verify the relationship between the GCD and LCM of 144, 180, and 192.
Solution:
The relation between GCD and LCM of 144, 180, and 192 is given as,
LCM(144, 180, 192) = [(144 × 180 × 192) × GCD(144, 180, 192)]/[GCD(144, 180) × GCD(180, 192) × GCD(144, 192)]
⇒ Prime factorization of 144, 180 and 192: 144 = 2^{4} × 3^{2}
 180 = 2^{2} × 3^{2} × 5^{1}
 192 = 2^{6} × 3^{1}
∴ GCD of (144, 180), (180, 192), (144, 192) and (144, 180, 192) = 36, 12, 48 and 12 respectively.
Now, LHS = LCM(144, 180, 192) = 2880.
And, RHS = [(144 × 180 × 192) × GCD(144, 180, 192)]/[GCD(144, 180) × GCD(180, 192) × GCD(144, 192)] = [(4976640) × 12]/[36 × 12 × 48] = 2880
LHS = RHS = 2880.
Hence verified. 
Example 3: Calculate the LCM of 144, 180, and 192 using the GCD of the given numbers.
Solution:
Prime factorization of 144, 180, 192:
 144 = 2^{4} × 3^{2}
 180 = 2^{2} × 3^{2} × 5^{1}
 192 = 2^{6} × 3^{1}
Therefore, GCD(144, 180) = 36, GCD(180, 192) = 12, GCD(144, 192) = 48, GCD(144, 180, 192) = 12
We know,
LCM(144, 180, 192) = [(144 × 180 × 192) × GCD(144, 180, 192)]/[GCD(144, 180) × GCD(180, 192) × GCD(144, 192)]
LCM(144, 180, 192) = (4976640 × 12)/(36 × 12 × 48) = 2880
⇒LCM(144, 180, 192) = 2880
FAQs on LCM of 144, 180, and 192
What is the LCM of 144, 180, and 192?
The LCM of 144, 180, and 192 is 2880. To find the LCM (least common multiple) of 144, 180, and 192, we need to find the multiples of 144, 180, and 192 (multiples of 144 = 144, 288, 432, 576 . . . . 2880 . . . . ; multiples of 180 = 180, 360, 540, 720 . . . . 2880 . . . . ; multiples of 192 = 192, 384, 576, 768 . . . . 2880 . . . . ) and choose the smallest multiple that is exactly divisible by 144, 180, and 192, i.e., 2880.
What are the Methods to Find LCM of 144, 180, 192?
The commonly used methods to find the LCM of 144, 180, 192 are:
 Division Method
 Prime Factorization Method
 Listing Multiples
What is the Relation Between GCF and LCM of 144, 180, 192?
The following equation can be used to express the relation between GCF and LCM of 144, 180, 192, i.e. LCM(144, 180, 192) = [(144 × 180 × 192) × GCF(144, 180, 192)]/[GCF(144, 180) × GCF(180, 192) × GCF(144, 192)].
How to Find the LCM of 144, 180, and 192 by Prime Factorization?
To find the LCM of 144, 180, and 192 using prime factorization, we will find the prime factors, (144 = 2^{4} × 3^{2}), (180 = 2^{2} × 3^{2} × 5^{1}), and (192 = 2^{6} × 3^{1}). LCM of 144, 180, and 192 is the product of prime factors raised to their respective highest exponent among the numbers 144, 180, and 192.
⇒ LCM of 144, 180, 192 = 2^{6} × 3^{2} × 5^{1} = 2880.
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