LCM of 8, 12, and 15
LCM of 8, 12, and 15 is the smallest number among all common multiples of 8, 12, and 15. The first few multiples of 8, 12, and 15 are (8, 16, 24, 32, 40 . . .), (12, 24, 36, 48, 60 . . .), and (15, 30, 45, 60, 75 . . .) respectively. There are 3 commonly used methods to find LCM of 8, 12, 15  by prime factorization, by division method, and by listing multiples.
1.  LCM of 8, 12, and 15 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 8, 12, and 15?
Answer: LCM of 8, 12, and 15 is 120.
Explanation:
The LCM of three nonzero integers, a(8), b(12), and c(15), is the smallest positive integer m(120) that is divisible by a(8), b(12), and c(15) without any remainder.
Methods to Find LCM of 8, 12, and 15
The methods to find the LCM of 8, 12, and 15 are explained below.
 By Prime Factorization Method
 By Division Method
 By Listing Multiples
LCM of 8, 12, and 15 by Prime Factorization
Prime factorization of 8, 12, and 15 is (2 × 2 × 2) = 2^{3}, (2 × 2 × 3) = 2^{2} × 3^{1}, and (3 × 5) = 3^{1} × 5^{1} respectively. LCM of 8, 12, and 15 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{3} × 3^{1} × 5^{1} = 120.
Hence, the LCM of 8, 12, and 15 by prime factorization is 120.
LCM of 8, 12, and 15 by Division Method
To calculate the LCM of 8, 12, and 15 by the division method, we will divide the numbers(8, 12, 15) by their prime factors (preferably common). The product of these divisors gives the LCM of 8, 12, and 15.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8, 12, and 15. Write this prime number(2) on the left of the given numbers(8, 12, and 15), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (8, 12, 15) 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 8, 12, and 15 is the product of all prime numbers on the left, i.e. LCM(8, 12, 15) by division method = 2 × 2 × 2 × 3 × 5 = 120.
LCM of 8, 12, and 15 by Listing Multiples
To calculate the LCM of 8, 12, 15 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 8 (8, 16, 24, 32, 40 . . .), 12 (12, 24, 36, 48, 60 . . .), and 15 (15, 30, 45, 60, 75 . . .).
 Step 2: The common multiples from the multiples of 8, 12, and 15 are 120, 240, . . .
 Step 3: The smallest common multiple of 8, 12, and 15 is 120.
∴ The least common multiple of 8, 12, and 15 = 120.
ā Also Check:
 LCM of 3 and 15  15
 LCM of 45 and 63  315
 LCM of 2, 3, 4, 5, 6 and 7  420
 LCM of 35 and 45  315
 LCM of 3, 7 and 10  210
 LCM of 30 and 36  180
 LCM of 20 and 36  180
LCM of 8, 12, and 15 Examples

Example 1: Verify the relationship between the GCD and LCM of 8, 12, and 15.
Solution:
The relation between GCD and LCM of 8, 12, and 15 is given as,
LCM(8, 12, 15) = [(8 × 12 × 15) × GCD(8, 12, 15)]/[GCD(8, 12) × GCD(12, 15) × GCD(8, 15)]
⇒ Prime factorization of 8, 12 and 15: 8 = 2^{3}
 12 = 2^{2} × 3^{1}
 15 = 3^{1} × 5^{1}
∴ GCD of (8, 12), (12, 15), (8, 15) and (8, 12, 15) = 4, 3, 1 and 1 respectively.
Now, LHS = LCM(8, 12, 15) = 120.
And, RHS = [(8 × 12 × 15) × GCD(8, 12, 15)]/[GCD(8, 12) × GCD(12, 15) × GCD(8, 15)] = [(1440) × 1]/[4 × 3 × 1] = 120
LHS = RHS = 120.
Hence verified. 
Example 2: Calculate the LCM of 8, 12, and 15 using the GCD of the given numbers.
Solution:
Prime factorization of 8, 12, 15:
 8 = 2^{3}
 12 = 2^{2} × 3^{1}
 15 = 3^{1} × 5^{1}
Therefore, GCD(8, 12) = 4, GCD(12, 15) = 3, GCD(8, 15) = 1, GCD(8, 12, 15) = 1
We know,
LCM(8, 12, 15) = [(8 × 12 × 15) × GCD(8, 12, 15)]/[GCD(8, 12) × GCD(12, 15) × GCD(8, 15)]
LCM(8, 12, 15) = (1440 × 1)/(4 × 3 × 1) = 120
⇒LCM(8, 12, 15) = 120 
Example 3: Find the smallest number that is divisible by 8, 12, 15 exactly.
Solution:
The smallest number that is divisible by 8, 12, and 15 exactly is their LCM.
⇒ Multiples of 8, 12, and 15: Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, . . . .
 Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . .
 Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, . . . .
Therefore, the LCM of 8, 12, and 15 is 120.
FAQs on LCM of 8, 12, and 15
What is the LCM of 8, 12, and 15?
The LCM of 8, 12, and 15 is 120. To find the LCM (least common multiple) of 8, 12, and 15, we need to find the multiples of 8, 12, and 15 (multiples of 8 = 8, 16, 24, 32 . . . . 120 . . . . ; multiples of 12 = 12, 24, 36, 48 . . . . 120 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 120 . . . . ) and choose the smallest multiple that is exactly divisible by 8, 12, and 15, i.e., 120.
What are the Methods to Find LCM of 8, 12, 15?
The commonly used methods to find the LCM of 8, 12, 15 are:
 Prime Factorization Method
 Listing Multiples
 Division Method
How to Find the LCM of 8, 12, and 15 by Prime Factorization?
To find the LCM of 8, 12, and 15 using prime factorization, we will find the prime factors, (8 = 2^{3}), (12 = 2^{2} × 3^{1}), and (15 = 3^{1} × 5^{1}). LCM of 8, 12, and 15 is the product of prime factors raised to their respective highest exponent among the numbers 8, 12, and 15.
⇒ LCM of 8, 12, 15 = 2^{3} × 3^{1} × 5^{1} = 120.
What is the Least Perfect Square Divisible by 8, 12, and 15?
The least number divisible by 8, 12, and 15 = LCM(8, 12, 15)
LCM of 8, 12, and 15 = 2 × 2 × 2 × 3 × 5 [Incomplete pair(s): 2, 3, 5]
⇒ Least perfect square divisible by each 8, 12, and 15 = LCM(8, 12, 15) × 2 × 3 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.
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