LCM of 7, 14, and 21
LCM of 7, 14, and 21 is the smallest number among all common multiples of 7, 14, and 21. The first few multiples of 7, 14, and 21 are (7, 14, 21, 28, 35 . . .), (14, 28, 42, 56, 70 . . .), and (21, 42, 63, 84, 105 . . .) respectively. There are 3 commonly used methods to find LCM of 7, 14, 21  by division method, by listing multiples, and by prime factorization.
1.  LCM of 7, 14, and 21 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 7, 14, and 21?
Answer: LCM of 7, 14, and 21 is 42.
Explanation:
The LCM of three nonzero integers, a(7), b(14), and c(21), is the smallest positive integer m(42) that is divisible by a(7), b(14), and c(21) without any remainder.
Methods to Find LCM of 7, 14, and 21
The methods to find the LCM of 7, 14, and 21 are explained below.
 By Listing Multiples
 By Prime Factorization Method
 By Division Method
LCM of 7, 14, and 21 by Listing Multiples
To calculate the LCM of 7, 14, 21 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 7 (7, 14, 21, 28, 35 . . .), 14 (14, 28, 42, 56, 70 . . .), and 21 (21, 42, 63, 84, 105 . . .).
 Step 2: The common multiples from the multiples of 7, 14, and 21 are 42, 84, . . .
 Step 3: The smallest common multiple of 7, 14, and 21 is 42.
∴ The least common multiple of 7, 14, and 21 = 42.
LCM of 7, 14, and 21 by Prime Factorization
Prime factorization of 7, 14, and 21 is (7) = 7^{1}, (2 × 7) = 2^{1} × 7^{1}, and (3 × 7) = 3^{1} × 7^{1} respectively. LCM of 7, 14, and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{1} × 3^{1} × 7^{1} = 42.
Hence, the LCM of 7, 14, and 21 by prime factorization is 42.
LCM of 7, 14, and 21 by Division Method
To calculate the LCM of 7, 14, and 21 by the division method, we will divide the numbers(7, 14, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 7, 14, and 21.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 7, 14, and 21. Write this prime number(2) on the left of the given numbers(7, 14, and 21), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (7, 14, 21) 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 7, 14, and 21 is the product of all prime numbers on the left, i.e. LCM(7, 14, 21) by division method = 2 × 3 × 7 = 42.
ā Also Check:
 LCM of 8 and 42  168
 LCM of 8, 9 and 12  72
 LCM of 8 and 56  56
 LCM of 25 and 36  900
 LCM of 2 and 13  26
 LCM of 8 and 18  72
 LCM of 8, 15 and 20  120
LCM of 7, 14, and 21 Examples

Example 1: Find the smallest number that is divisible by 7, 14, 21 exactly.
Solution:
The smallest number that is divisible by 7, 14, and 21 exactly is their LCM.
⇒ Multiples of 7, 14, and 21: Multiples of 7 = 7, 14, 21, 28, 35, 42, . . . .
 Multiples of 14 = 14, 28, 42, 56, 70, 84, . . . .
 Multiples of 21 = 21, 42, 63, 84, 105, 126, . . . .
Therefore, the LCM of 7, 14, and 21 is 42.

Example 2: Calculate the LCM of 7, 14, and 21 using the GCD of the given numbers.
Solution:
Prime factorization of 7, 14, 21:
 7 = 7^{1}
 14 = 2^{1} × 7^{1}
 21 = 3^{1} × 7^{1}
Therefore, GCD(7, 14) = 7, GCD(14, 21) = 7, GCD(7, 21) = 7, GCD(7, 14, 21) = 7
We know,
LCM(7, 14, 21) = [(7 × 14 × 21) × GCD(7, 14, 21)]/[GCD(7, 14) × GCD(14, 21) × GCD(7, 21)]
LCM(7, 14, 21) = (2058 × 7)/(7 × 7 × 7) = 42
⇒LCM(7, 14, 21) = 42 
Example 3: Verify the relationship between the GCD and LCM of 7, 14, and 21.
Solution:
The relation between GCD and LCM of 7, 14, and 21 is given as,
LCM(7, 14, 21) = [(7 × 14 × 21) × GCD(7, 14, 21)]/[GCD(7, 14) × GCD(14, 21) × GCD(7, 21)]
⇒ Prime factorization of 7, 14 and 21: 7 = 7^{1}
 14 = 2^{1} × 7^{1}
 21 = 3^{1} × 7^{1}
∴ GCD of (7, 14), (14, 21), (7, 21) and (7, 14, 21) = 7, 7, 7 and 7 respectively.
Now, LHS = LCM(7, 14, 21) = 42.
And, RHS = [(7 × 14 × 21) × GCD(7, 14, 21)]/[GCD(7, 14) × GCD(14, 21) × GCD(7, 21)] = [(2058) × 7]/[7 × 7 × 7] = 42
LHS = RHS = 42.
Hence verified.
FAQs on LCM of 7, 14, and 21
What is the LCM of 7, 14, and 21?
The LCM of 7, 14, and 21 is 42. To find the least common multiple of 7, 14, and 21, we need to find the multiples of 7, 14, and 21 (multiples of 7 = 7, 14, 21, 28, 42 . . . .; multiples of 14 = 14, 28, 42, 56 . . . .; multiples of 21 = 21, 42, 63, 84 . . . .) and choose the smallest multiple that is exactly divisible by 7, 14, and 21, i.e., 42.
What are the Methods to Find LCM of 7, 14, 21?
The commonly used methods to find the LCM of 7, 14, 21 are:
 Listing Multiples
 Prime Factorization Method
 Division Method
What is the Least Perfect Square Divisible by 7, 14, and 21?
The least number divisible by 7, 14, and 21 = LCM(7, 14, 21)
LCM of 7, 14, and 21 = 2 × 3 × 7 [Incomplete pair(s): 2, 3, 7]
⇒ Least perfect square divisible by each 7, 14, and 21 = LCM(7, 14, 21) × 2 × 3 × 7 = 1764 [Square root of 1764 = √1764 = ±42]
Therefore, 1764 is the required number.
How to Find the LCM of 7, 14, and 21 by Prime Factorization?
To find the LCM of 7, 14, and 21 using prime factorization, we will find the prime factors, (7 = 7^{1}), (14 = 2^{1} × 7^{1}), and (21 = 3^{1} × 7^{1}). LCM of 7, 14, and 21 is the product of prime factors raised to their respective highest exponent among the numbers 7, 14, and 21.
⇒ LCM of 7, 14, 21 = 2^{1} × 3^{1} × 7^{1} = 42.
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