LCM of 9 and 21
LCM of 9 and 21 is the smallest number among all common multiples of 9 and 21. The first few multiples of 9 and 21 are (9, 18, 27, 36, 45, 54, . . . ) and (21, 42, 63, 84, 105, 126, 147, . . . ) respectively. There are 3 commonly used methods to find LCM of 9 and 21  by listing multiples, by prime factorization, and by division method.
1.  LCM of 9 and 21 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 9 and 21?
Answer: LCM of 9 and 21 is 63.
Explanation:
The LCM of two nonzero integers, x(9) and y(21), is the smallest positive integer m(63) that is divisible by both x(9) and y(21) without any remainder.
Methods to Find LCM of 9 and 21
The methods to find the LCM of 9 and 21 are explained below.
 By Division Method
 By Listing Multiples
 By Prime Factorization Method
LCM of 9 and 21 by Division Method
To calculate the LCM of 9 and 21 by the division method, we will divide the numbers(9, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 9 and 21.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 9 and 21. Write this prime number(3) on the left of the given numbers(9 and 21), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (9, 21) is a multiple of 3, divide it by 3 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 9 and 21 is the product of all prime numbers on the left, i.e. LCM(9, 21) by division method = 3 × 3 × 7 = 63.
LCM of 9 and 21 by Listing Multiples
To calculate the LCM of 9 and 21 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 9 (9, 18, 27, 36, 45, 54, . . . ) and 21 (21, 42, 63, 84, 105, 126, 147, . . . . )
 Step 2: The common multiples from the multiples of 9 and 21 are 63, 126, . . .
 Step 3: The smallest common multiple of 9 and 21 is 63.
∴ The least common multiple of 9 and 21 = 63.
LCM of 9 and 21 by Prime Factorization
Prime factorization of 9 and 21 is (3 × 3) = 3^{2} and (3 × 7) = 3^{1} × 7^{1} respectively. LCM of 9 and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 3^{2} × 7^{1} = 63.
Hence, the LCM of 9 and 21 by prime factorization is 63.
☛ Also Check:
 LCM of 24, 36 and 48  144
 LCM of 10 and 30  30
 LCM of 30, 45 and 60  180
 LCM of 8 and 16  16
 LCM of 9, 12 and 15  180
 LCM of 24 and 36  72
 LCM of 2 and 15  30
LCM of 9 and 21 Examples

Example 1: The GCD and LCM of two numbers are 3 and 63 respectively. If one number is 21, find the other number.
Solution:
Let the other number be a.
∵ GCD × LCM = 21 × a
⇒ a = (GCD × LCM)/21
⇒ a = (3 × 63)/21
⇒ a = 9
Therefore, the other number is 9. 
Example 2: The product of two numbers is 189. If their GCD is 3, what is their LCM?
Solution:
Given: GCD = 3
product of numbers = 189
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 189/3
Therefore, the LCM is 63.
The probable combination for the given case is LCM(9, 21) = 63. 
Example 3: Verify the relationship between GCF and LCM of 9 and 21.
Solution:
The relation between GCF and LCM of 9 and 21 is given as,
LCM(9, 21) × GCF(9, 21) = Product of 9, 21
Prime factorization of 9 and 21 is given as, 9 = (3 × 3) = 3^{2} and 21 = (3 × 7) = 3^{1} × 7^{1}
LCM(9, 21) = 63
GCF(9, 21) = 3
LHS = LCM(9, 21) × GCF(9, 21) = 63 × 3 = 189
RHS = Product of 9, 21 = 9 × 21 = 189
⇒ LHS = RHS = 189
Hence, verified.
FAQs on LCM of 9 and 21
What is the LCM of 9 and 21?
The LCM of 9 and 21 is 63. To find the least common multiple (LCM) of 9 and 21, we need to find the multiples of 9 and 21 (multiples of 9 = 9, 18, 27, 36 . . . . 63; multiples of 21 = 21, 42, 63, 84) and choose the smallest multiple that is exactly divisible by 9 and 21, i.e., 63.
How to Find the LCM of 9 and 21 by Prime Factorization?
To find the LCM of 9 and 21 using prime factorization, we will find the prime factors, (9 = 3 × 3) and (21 = 3 × 7). LCM of 9 and 21 is the product of prime factors raised to their respective highest exponent among the numbers 9 and 21.
⇒ LCM of 9, 21 = 3^{2} × 7^{1} = 63.
What is the Least Perfect Square Divisible by 9 and 21?
The least number divisible by 9 and 21 = LCM(9, 21)
LCM of 9 and 21 = 3 × 3 × 7 [Incomplete pair(s): 7]
⇒ Least perfect square divisible by each 9 and 21 = LCM(9, 21) × 7 = 441 [Square root of 441 = √441 = ±21]
Therefore, 441 is the required number.
Which of the following is the LCM of 9 and 21? 63, 35, 11, 10
The value of LCM of 9, 21 is the smallest common multiple of 9 and 21. The number satisfying the given condition is 63.
If the LCM of 21 and 9 is 63, Find its GCF.
LCM(21, 9) × GCF(21, 9) = 21 × 9
Since the LCM of 21 and 9 = 63
⇒ 63 × GCF(21, 9) = 189
Therefore, the greatest common factor = 189/63 = 3.
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