LCM of 6, 72, and 120
LCM of 6, 72, and 120 is the smallest number among all common multiples of 6, 72, and 120. The first few multiples of 6, 72, and 120 are (6, 12, 18, 24, 30 . . .), (72, 144, 216, 288, 360 . . .), and (120, 240, 360, 480, 600 . . .) respectively. There are 3 commonly used methods to find LCM of 6, 72, 120  by prime factorization, by listing multiples, and by division method.
1.  LCM of 6, 72, and 120 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 6, 72, and 120?
Answer: LCM of 6, 72, and 120 is 360.
Explanation:
The LCM of three nonzero integers, a(6), b(72), and c(120), is the smallest positive integer m(360) that is divisible by a(6), b(72), and c(120) without any remainder.
Methods to Find LCM of 6, 72, and 120
Let's look at the different methods for finding the LCM of 6, 72, and 120.
 By Division Method
 By Prime Factorization Method
 By Listing Multiples
LCM of 6, 72, and 120 by Division Method
To calculate the LCM of 6, 72, and 120 by the division method, we will divide the numbers(6, 72, 120) by their prime factors (preferably common). The product of these divisors gives the LCM of 6, 72, and 120.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6, 72, and 120. Write this prime number(2) on the left of the given numbers(6, 72, and 120), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (6, 72, 120) 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 6, 72, and 120 is the product of all prime numbers on the left, i.e. LCM(6, 72, 120) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.
LCM of 6, 72, and 120 by Prime Factorization
Prime factorization of 6, 72, and 120 is (2 × 3) = 2^{1} × 3^{1}, (2 × 2 × 2 × 3 × 3) = 2^{3} × 3^{2}, and (2 × 2 × 2 × 3 × 5) = 2^{3} × 3^{1} × 5^{1} respectively. LCM of 6, 72, and 120 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{3} × 3^{2} × 5^{1} = 360.
Hence, the LCM of 6, 72, and 120 by prime factorization is 360.
LCM of 6, 72, and 120 by Listing Multiples
To calculate the LCM of 6, 72, 120 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 6 (6, 12, 18, 24, 30 . . .), 72 (72, 144, 216, 288, 360 . . .), and 120 (120, 240, 360, 480, 600 . . .).
 Step 2: The common multiples from the multiples of 6, 72, and 120 are 360, 720, . . .
 Step 3: The smallest common multiple of 6, 72, and 120 is 360.
∴ The least common multiple of 6, 72, and 120 = 360.
ā Also Check:
 LCM of 18 and 28  252
 LCM of 8, 15 and 20  120
 LCM of 5 and 24  120
 LCM of 56 and 72  504
 LCM of 6, 7 and 9  126
 LCM of 6 and 21  42
 LCM of 8 and 64  64
LCM of 6, 72, and 120 Examples

Example 1: Calculate the LCM of 6, 72, and 120 using the GCD of the given numbers.
Solution:
Prime factorization of 6, 72, 120:
 6 = 2^{1} × 3^{1}
 72 = 2^{3} × 3^{2}
 120 = 2^{3} × 3^{1} × 5^{1}
Therefore, GCD(6, 72) = 6, GCD(72, 120) = 24, GCD(6, 120) = 6, GCD(6, 72, 120) = 6
We know,
LCM(6, 72, 120) = [(6 × 72 × 120) × GCD(6, 72, 120)]/[GCD(6, 72) × GCD(72, 120) × GCD(6, 120)]
LCM(6, 72, 120) = (51840 × 6)/(6 × 24 × 6) = 360
⇒LCM(6, 72, 120) = 360 
Example 2: Verify the relationship between the GCD and LCM of 6, 72, and 120.
Solution:
The relation between GCD and LCM of 6, 72, and 120 is given as,
LCM(6, 72, 120) = [(6 × 72 × 120) × GCD(6, 72, 120)]/[GCD(6, 72) × GCD(72, 120) × GCD(6, 120)]
⇒ Prime factorization of 6, 72 and 120: 6 = 2^{1} × 3^{1}
 72 = 2^{3} × 3^{2}
 120 = 2^{3} × 3^{1} × 5^{1}
∴ GCD of (6, 72), (72, 120), (6, 120) and (6, 72, 120) = 6, 24, 6 and 6 respectively.
Now, LHS = LCM(6, 72, 120) = 360.
And, RHS = [(6 × 72 × 120) × GCD(6, 72, 120)]/[GCD(6, 72) × GCD(72, 120) × GCD(6, 120)] = [(51840) × 6]/[6 × 24 × 6] = 360
LHS = RHS = 360.
Hence verified. 
Example 3: Find the smallest number that is divisible by 6, 72, 120 exactly.
Solution:
The value of LCM(6, 72, 120) will be the smallest number that is exactly divisible by 6, 72, and 120.
⇒ Multiples of 6, 72, and 120: Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . . ., 336, 342, 348, 354, 360, . . . .
 Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, 576, 648, 720, . . . ., 144, 216, 288, 360, . . . .
 Multiples of 120 = 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, . . . ., 120, 0, 120, 240, 360, . . . .
Therefore, the LCM of 6, 72, and 120 is 360.
FAQs on LCM of 6, 72, and 120
What is the LCM of 6, 72, and 120?
The LCM of 6, 72, and 120 is 360. To find the LCM of 6, 72, and 120, we need to find the multiples of 6, 72, and 120 (multiples of 6 = 6, 12, 18, 24 . . . . 360 . . . . ; multiples of 72 = 72, 144, 216, 288 . . . . 360 . . . . ; multiples of 120 = 120, 240, 360, 480 . . . .) and choose the smallest multiple that is exactly divisible by 6, 72, and 120, i.e., 360.
Which of the following is the LCM of 6, 72, and 120? 360, 21, 110, 3
The value of LCM of 6, 72, 120 is the smallest common multiple of 6, 72, and 120. The number satisfying the given condition is 360.
What are the Methods to Find LCM of 6, 72, 120?
The commonly used methods to find the LCM of 6, 72, 120 are:
 Division Method
 Listing Multiples
 Prime Factorization Method
What is the Relation Between GCF and LCM of 6, 72, 120?
The following equation can be used to express the relation between GCF and LCM of 6, 72, 120, i.e. LCM(6, 72, 120) = [(6 × 72 × 120) × GCF(6, 72, 120)]/[GCF(6, 72) × GCF(72, 120) × GCF(6, 120)].
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