LCM of 5, 6, and 7
LCM of 5, 6, and 7 is the smallest number among all common multiples of 5, 6, and 7. The first few multiples of 5, 6, and 7 are (5, 10, 15, 20, 25 . . .), (6, 12, 18, 24, 30 . . .), and (7, 14, 21, 28, 35 . . .) respectively. There are 3 commonly used methods to find LCM of 5, 6, 7  by listing multiples, by division method, and by prime factorization.
1.  LCM of 5, 6, and 7 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 5, 6, and 7?
Answer: LCM of 5, 6, and 7 is 210.
Explanation:
The LCM of three nonzero integers, a(5), b(6), and c(7), is the smallest positive integer m(210) that is divisible by a(5), b(6), and c(7) without any remainder.
Methods to Find LCM of 5, 6, and 7
The methods to find the LCM of 5, 6, and 7 are explained below.
 By Prime Factorization Method
 By Division Method
 By Listing Multiples
LCM of 5, 6, and 7 by Prime Factorization
Prime factorization of 5, 6, and 7 is (5) = 5^{1}, (2 × 3) = 2^{1} × 3^{1}, and (7) = 7^{1} respectively. LCM of 5, 6, and 7 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{1} × 3^{1} × 5^{1} × 7^{1} = 210.
Hence, the LCM of 5, 6, and 7 by prime factorization is 210.
LCM of 5, 6, and 7 by Division Method
To calculate the LCM of 5, 6, and 7 by the division method, we will divide the numbers(5, 6, 7) by their prime factors (preferably common). The product of these divisors gives the LCM of 5, 6, and 7.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 5, 6, and 7. Write this prime number(2) on the left of the given numbers(5, 6, and 7), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (5, 6, 7) 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 5, 6, and 7 is the product of all prime numbers on the left, i.e. LCM(5, 6, 7) by division method = 2 × 3 × 5 × 7 = 210.
LCM of 5, 6, and 7 by Listing Multiples
To calculate the LCM of 5, 6, 7 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 5 (5, 10, 15, 20, 25 . . .), 6 (6, 12, 18, 24, 30 . . .), and 7 (7, 14, 21, 28, 35 . . .).
 Step 2: The common multiples from the multiples of 5, 6, and 7 are 210, 420, . . .
 Step 3: The smallest common multiple of 5, 6, and 7 is 210.
∴ The least common multiple of 5, 6, and 7 = 210.
ā Also Check:
 LCM of 14 and 20  140
 LCM of 144 and 169  24336
 LCM of 39 and 65  195
 LCM of 10 and 15  30
 LCM of 6, 8 and 12  24
 LCM of 5 and 14  70
 LCM of 8, 12 and 24  24
LCM of 5, 6, and 7 Examples

Example 1: Find the smallest number that is divisible by 5, 6, 7 exactly.
Solution:
The value of LCM(5, 6, 7) will be the smallest number that is exactly divisible by 5, 6, and 7.
⇒ Multiples of 5, 6, and 7: Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, . . . ., 200, 205, 210, . . . .
 Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . . ., 192, 198, 204, 210, . . . .
 Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, . . . ., 189, 196, 203, 210, . . . .
Therefore, the LCM of 5, 6, and 7 is 210.

Example 2: Verify the relationship between the GCD and LCM of 5, 6, and 7.
Solution:
The relation between GCD and LCM of 5, 6, and 7 is given as,
LCM(5, 6, 7) = [(5 × 6 × 7) × GCD(5, 6, 7)]/[GCD(5, 6) × GCD(6, 7) × GCD(5, 7)]
⇒ Prime factorization of 5, 6 and 7: 5 = 5^{1}
 6 = 2^{1} × 3^{1}
 7 = 7^{1}
∴ GCD of (5, 6), (6, 7), (5, 7) and (5, 6, 7) = 1, 1, 1 and 1 respectively.
Now, LHS = LCM(5, 6, 7) = 210.
And, RHS = [(5 × 6 × 7) × GCD(5, 6, 7)]/[GCD(5, 6) × GCD(6, 7) × GCD(5, 7)] = [(210) × 1]/[1 × 1 × 1] = 210
LHS = RHS = 210.
Hence verified. 
Example 3: Calculate the LCM of 5, 6, and 7 using the GCD of the given numbers.
Solution:
Prime factorization of 5, 6, 7:
 5 = 5^{1}
 6 = 2^{1} × 3^{1}
 7 = 7^{1}
Therefore, GCD(5, 6) = 1, GCD(6, 7) = 1, GCD(5, 7) = 1, GCD(5, 6, 7) = 1
We know,
LCM(5, 6, 7) = [(5 × 6 × 7) × GCD(5, 6, 7)]/[GCD(5, 6) × GCD(6, 7) × GCD(5, 7)]
LCM(5, 6, 7) = (210 × 1)/(1 × 1 × 1) = 210
⇒LCM(5, 6, 7) = 210
FAQs on LCM of 5, 6, and 7
What is the LCM of 5, 6, and 7?
The LCM of 5, 6, and 7 is 210. To find the LCM of 5, 6, and 7, we need to find the multiples of 5, 6, and 7 (multiples of 5 = 5, 10, 15, 20 . . . . 210 . . . . ; multiples of 6 = 6, 12, 18, 24 . . . . 210 . . . . ; multiples of 7 = 7, 14, 21, 28 . . . . 210 . . . . ) and choose the smallest multiple that is exactly divisible by 5, 6, and 7, i.e., 210.
What is the Relation Between GCF and LCM of 5, 6, 7?
The following equation can be used to express the relation between GCF and LCM of 5, 6, 7, i.e. LCM(5, 6, 7) = [(5 × 6 × 7) × GCF(5, 6, 7)]/[GCF(5, 6) × GCF(6, 7) × GCF(5, 7)].
What is the Least Perfect Square Divisible by 5, 6, and 7?
The least number divisible by 5, 6, and 7 = LCM(5, 6, 7)
LCM of 5, 6, and 7 = 2 × 3 × 5 × 7 [Incomplete pair(s): 2, 3, 5, 7]
⇒ Least perfect square divisible by each 5, 6, and 7 = LCM(5, 6, 7) × 2 × 3 × 5 × 7 = 44100 [Square root of 44100 = √44100 = ±210]
Therefore, 44100 is the required number.
What are the Methods to Find LCM of 5, 6, 7?
The commonly used methods to find the LCM of 5, 6, 7 are:
 Prime Factorization Method
 Listing Multiples
 Division Method
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